text:
Facilitating the Spread of Knowledge and Innovation in Professional Software Development

Write for InfoQ

### Choose your language

InfoQ Homepage News LLVM Proposes Adding Modules to C

# LLVM Proposes Adding Modules to C

This item in japanese

At the November LLVM developers meeting, Doug Gregor of Apple gave a presentation on adding modules to C. From the talk's abstract:

The C preprocessor has long been a source of problems for programmers and tools alike. Programmers must contend with widespread macro pollution and include-ordering problems due to ill-behaved headers. Developers habitually employ various preprocessor workarounds, such as LONG_MACRO_PREFIXES, include guards, and the occasional #undef of a library macro to mitigate these problems.

Tools, on the other hand, must cope with the inherent scalability problems associated with parsing the same headers repeatedly, because each different preprocessing context could effect how a header is interpreted---even though the programmer rarely wants it.

Modules seeks to solve this problem by isolating the interface of a particular library and compiling it (once) into an efficient, serialized representation that can be efficiently imported whenever that library is used, improving both the programmer's experience and the scalability of the compilation process.

The basic premise of the proposal is to avoid having to use the pre-processor to include vast quantities of headers when compiling even the simplest files, as a way of speeding up compilation and allowing for re-use of previously parsed headers. In the eponymous "Hello World" example, he highlights how a 64-character C program becomes 11,074 characters after pre-procesing, and an 81-character C++ program becomes 1,161,033 characters after pre-processing. He also notes that re-parsing headers can lead to fragility, since the inclusion depends on the state of the pre-processor at the time (for example, with #define FILE "myfile.txt" before #include <stdio.h>, the pre-processor will mangle the headers and result in a failed build).

The proposal is to use a new keyword, import, to load the module. Instead of being a pre-processor textual inclusion, the compiler can understand that this is a fixed version and thus only parse the module once. Additional uses of the same module can use the same previously parsed data structures instead of starting anew each time.

Modules can also be nested, which allows a sub-module to be imported; in the example given, he demonstrates a std module with a stdio submodule, leading to inclusion with import std.stdio. Importing a module brings in all its public API to the client but can hide its non-public APIs. For that to happen, the module needs to declare what is and isn't public, for which the public keyword can be used:

// stdio.c
export std.stdio:
public:
typedef struct {
…
} FILE;
int printf(const char*, …) {
…
}

Note that in this example, it is sufficient to just provide the implementation of the file – no headers are needed. The export contains the name of the module (std.stdio in this case) and the public demarcates the public vs non-public parts of the API. This can be compiled into both the library and sufficient metadata for the function types and macros to be available in client code.

Of course, this is a suggestion for the future and not a standard, so how will it happen? The proposal is to use headers for the public API of existing modules, and define modules to be a set of headers:

// /usr/include/module.map
module std {
module stdio { header "stdio.h" }
module stdlib { header "stdlib.h" }
module math { header "math.h" }
}
module ClangAST {
umbrella "AST/AST.h"
module * { }
} // can use "import ClangAST.Decl" for AST/Decl.h

To facilitate module generation in future (and in part to make it easy for Objective-C frameworks to export modules) an 'umbrella module' allows a set of headers in a directory to be exported as a single module.

Compilers that are adapted to process modules can take advantage of a single pass over the headers to construct the module, and then re-use that module information in subsequent headers. (The format of the compiled module isn't yet specified, and may be compiler-specific.) Modules may also add additional meta-information, such as what libraries are required for the module to run – this will allow the compiler to pick up on the linking flags needed for each of the modules without the user having to build up a large array of -l flags at link time.

To take advantage of modules, the only change required at the client side will be to replace the #include with an equivalent import. Additionally, modules can provide better compiler diagnostics since they have the information of what modules export which functions/types after pre-processing is done; this will allow compiler errors and IDE quick fixes to suggest adding the required import instead of falling over.

Finally, re-using module information also permits debugging information to be associated with the module, instead of with every object file that replicates the contents. This results in both the compiler and linker emitting less debugging information, which in turn, speeds up the compilation time. It also provides additional type information to the debugger, so the debugger will report the correct types as defined in the module (instead of them being the result of in-lined text in each object file).

The net effect of the modules proposal is to provide a transition path and compatibility with existing tools, whilst providing benefits (primarily in the form of compilation speed and improved diagnostic error messages/debugging) with very little change required by the users. It also provides an incremental way that files can be upgraded, switching from individual pre-processor instructions to module based imports. Whilst no compilation speed measurements were given as part of the modules presentation, work is already underway in LLVM to implement these modules. And although the modules don't take into account either versioning or name-spacing (largely due to having to build in backward compatibility) the benefits, if widely adopted, will have a significant boost in compilation speed of C and C++ based programs. Furthermore, the backward compatibility story has been explicitly laid out, and like the LLVM blocks specification, is likely to be made available for inclusion in other compilers or standards as and when required. However, of the widely used C and C++ compilers, the LLVM compiler toolchain has been the only one innovating and leading by example. Whether these features are introduced in other compilers will likely depend on the success of the LLVM implementation and the benefits it confers.

Style

## Hello stranger!

You need to Register an InfoQ account or or login to post comments. But there's so much more behind being registered.

Get the most out of the InfoQ experience.

Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p


text:
INSECT (IN-silico SEarch for Co-occurring Transcription factors) is a user-friendly tool to predict the occurrence of Cis Regulatory Modules (CRMs), which control gene expression. INSECT allows a complete and flexible analysis of the predicted co-regulating Transcription Factors (TFs) and Transcription Factor Binding Sites (TFBSs). The tool combines several CRM identification strategies by applying different search schemes and restriction parameters on TFs, TFBSs and genome sequences. CRMs search is performed by using Positional Weight Matrices (PWM), which are publicly available in TRANSFAC, JASPAR, UniPROBE, SwissRegulon, Hocomoco and FlyFactorSurvey databases, or by constructing PWMs from known target sequences. INSECT integrates many different search options and additional results such as automatic regulatory sequences retrieval from Ensembl, phylogenetic footprinting, nucleosome occupancy calculations, and gene ontology information. INSECT output includes identified CRM site location graphics and statistics on potential TFs positions and PWM Scores. CRMs site information can be uploaded to UCSC Genome Browser as GFF file tracks to be compared and integrated with other relevant genomic information such as data from the ENCODE project. The simple and intuitive interface along with its search flexibility and data integration makes INSECT an easy and powerful tool for scientists requiring search and analysis capabilities for potential CRM presence at a genomic scale.

#### Citation

If you find INSECT 2.0 useful for your work, please cite us:

Parra RG, Rohr CO, Koile D, Perez-Castro C and Yankilevich P.
INSECT 2.0: a web-server for genome-wide cis-regulatory modules prediction.
Bioinformatics 2016 32:1229-1231.

## 1. Cis-regulatory modules

The rate of transcription of genetic information from DNA to mRNA is controlled by transcription factors proteins that bind to specific DNA sequences named transcription factor binding sites. These proteins perform their regulatory function alone or with other proteins in a complex, by promoting (as an activator), or blocking (as a repressor) the recruitment of RNA polymerase to specific genes. Some transcription factors interactions tend to co-localize at specific DNA regions, usually 100-1000 DNA base pairs in length, named Cis-regulatory modules.

 Figure 1. Transcription factors bind to specific DNA regions named transcription factor binding sites. Sets of TFs can interact conforming Cis-regulatory modules. Nat Rev Genet, Vol. 5, No. 4. (01 April 2004), pp. 276-287.

## 2. Analysis type

### 2.1 Single organism

In single organism searches: whole-genome, lists of genes or fasta sequences are analysed for the presence of TFBSs of interest which conform a potential CRM (see Search section to see the methodology).

### 2.2 Phylogenetic footprinting

Phylogenetic footprinting is a technique used to identify transcription factor binding sites (TFBS) within a non-coding region of DNA of interest by comparing it to the orthologous sequence in different species. INSECT uses this technique to lower the false positives rate in TFBS searches. For every potential CRM identified in a gene, the INSECT search algorithm requires the orthologous gene sequence to have an equivalent CRM. A CRM is considered equivalent in a pair of orthologous genes if the TFBS positions relative to the gene Transcription Start Site (TSS) are conserved within a range defined by the user (by default it is set to 1000 bp). Users are encouraged to carefully apply this option as it has been shown that between human and mouse ~50% of regulatory elements are conserved. It has been also postulated that most of the genome conserved sites are those of high binding specificity. Note that filtering out a specific site by applying phylogenetic footprinting does not necessarily mean that it is a false positive; instead, it means that the other remaining sites have more chances to be true positives due to a high conservation.

## 3. Search methods

CRMs are defined by a set of TFBS and specific relations among them. The INSECT web server allows users to perform two different TFBS searching methods.

### 3.1 Sliding window search

In the sliding window method, the genome or target sequences are scanned using a fixed length sliding window defined by the user. As the windows slides the analysed subsequences must contain matches to all the selected TFs in order to report a potential CRM. Additionally, users can set some of the TFs not be necessarily present to report a valid CRM by checking the “Allow missing” option located in the “Transcription Factors and TFBSs” section alongside the selected TFs.

### 3.2 Master driven search

In the master driven method a master transcription factor must be selected to drive the search of the co-occurring factors, which are restricted to be located at a user defined spacing relative to the master. This method is more restrictive than the sliding window search as it requires the different TFs to strictly satisfy the imposed restrictions in relation to a reference master TFBS. A maximum of 5 TFBSs can be search in this mode, including the master.

Note: For both search options TFBSs can be searched either in the direct, reverse or both strand orientations. This is an important feature as usually TFBSs co-localize in very specific manners when binding to DNA as a consequence of their physical interaction at their quaternary structure.

## 4. Input parameters

### 4.1 Input sequences

INSECT allows automatic sequence retrieval from a group of fourteen genomes that are deposited in the Ensembl genome database (www.ensembl.org). These genomes were downloaded to our INSECT server, and were updated to their last version according to the Ensembl 2015 release.

#### 4.1.1 Automatic sequence retrieval from a gene list

Two different types of IDs are accepted for automatic sequence retrieval: Gene Symbols (e.g. BRCA2) or Ensembl Gene IDs (e.g. ENSG00000139618). These IDs can be typed in a text area, separated by carriage returns.

#### 4.1.2 Automatic sequence retrieval from a whole genome

INSECT allows performing CRM searches in whole genomes for the fourteen Ensembl genomes available in our database. By using this option, INSECT will automatically search all the annotated genes from Ensembl for the selected genome.

Note on gene filters: Some researchers might be interested to analyse specific subsets of genes from the genome, i.e. a specific chromosome or genes from a given biotype. INSECT allows the user to apply different input data filters to reduce the analysis to the specified genomic elements, which will help to reduce the search space and save time to focus on the dataset of interest.

#### 4.1.3 User-defined sequences

Users can upload their own set of sequences to perform the CRM search. These sequences must be stored in a multi-fasta file properly formatted. Follow the link to see a valid Multi-Fasta file format.

### 4.2 Distance from the TSS

The automatic sequence retrieval requires the users to provide a valid list of Gene IDs and the upstream and downstream distance limit of the putative regulatory regions relative to the transcription start site. Taking these distances into account INSECT retrieves the sequences to analyse for the whole gene list.

### 4.3 Orthologous conservation window size

In phylogenetic footprinting analysis a potential CRMs need to be conserved in both species. The CRM must be located within a certain window size when comparing positions relative to the corresponding TSS of each specie (e.g. if the conservation window size is of 2000 bp, and there is a CRM located 1500 bp upstream from the reference gene, the CRM must be found in-between the positions 500 bp and 2500 bp upstream in the orthologous sequence in order to constitute a valid site).

### 4.4 PWMs

A position weight matrix (PWM), also called position-specific weight matrix (PSWM) or position-specific scoring matrix (PSSM), is a commonly used representation of motifs (patterns) in biological sequences. A PWM is a matrix of score values that gives a weighted match to any given substring of a fixed length within a sequence. It has one row for each symbol of the alphabet, and one column for each position in the pattern. PWMs representing different transcription factor binding sites (TFBSs) can be selected from the INSECT local database where JASPAR (JASPAR_CORE), TRANSFAC Release 7.0 (Public), UniPROBE, SwissRegulon, Hocomoco and FlyFactorSurvey PWMs are integrated. Alternatively users can upload their own PWMs or create them from an ungapped multiple sequence alignment (msa).

PWM calculation from a msa:

From a set S of n aligned sequences of length l, s1, ... , sn, where sk = sk1, ... , skl (the skj being one of {A, C, G, T}) a Position Weight Matrix, M4xl is derived as $M_{ij} = _{n}^{1}\sum_{k=1}^{n} I_{i} (s_{kj})$ where $i = A,C,G,T$ and $j=1,...,l$
where $I_{i}(q) = \left\{\begin{matrix} 1\ if \ i=q\\ 0 \ otherwise \end{matrix}\right.$
Each coefficient in this matrix indicates the number of times that a given nucleotide has been observed at a given position. The coefficients in a Position Weight Matrix are directly computed as log-likelyhood values according with the following transformation log( Mij/pi), where Mij is the probability of nucleotide i at position j in the Matrix M, and pi is the background probability of nucleotide i. The background probability of nucleotide can be assumed to be an 'a priory' probability, the frequency of the nucleotide in the whole sequences use to derive the matrix, or the frequency in the aligned region from where the matrix is actually derived. Then, given a sequence of length l above log-likelihood ratio can be computed by adding the coefficients of the log-likelihood matrix corresponding to each nucleotide in each position on the sequence.

### 4.5 PWM Scoring and threshold setting

The score assigned by a PWM $m$ to a substring $S=(S{j})_{j=1}^{N}$ is defined as $\sum_{j=1}^{N} m_{s_{j},j}$, where $j$ represents position in the substring, $s_{j}$ is the symbol at position $j$ in the substring, and $m_{\alpha ,j}$ is the score in row $\alpha$, column $j$ of the matrix. PWM scores are calculated as the sum of position-specific scores for each symbol in a given substring. We assume a direct association between the DNA sequence variability in the binding sites and the binding afﬁnity (or activity) for the particular protein that recognizes these sites. In this context, if enough number of sequences were employed to build the PWM, higher TF binding afﬁnities would be indicated by higher PWM scores, being the maximum matrix score, $MMSc = \sum_{j=1}^{N} max(m_{j})$. Substrings will have PWM scores no greater than its MMSc. The score of every substring is divided by the MMSc in order to normalize the TF afﬁnities and provide a score that is comparable among different factors without a length motif bias.
TFBSs search using a PWM requires defining a score threshold in order to determine if a sequence is defined as a potential TFBS or a false positive. The threshold value is expressed as a percentage. A low score threshold usually leads to the appearance of spurious matches, making the prediction of potential TFBSs difficult due to a high False Positive Rate (FPR). Therefore, users need to define and set an optimal threshold for each search.

### 4.6 TFBSs search parameters

#### 4.6.1 Sliding window parameters

##### 4.6.1.1 TFs Score Threshold
A threshold value to be consider for search of all the TFs occurrences. See PWM Scoring and threshold setting for an explanation.
##### 4.6.1.2 Co-occurrence window size
This is the size of the window where the co-occurring factors are supposed to be located. The algorithm works by sliding a window along the target sequence and checking for the presence of every TF that was added to the search.

 Figure 2. Sliding window search for three co-occurring TFBSs with spacing restriction set by a window size of N bp.
##### 4.6.1.3 Co-occurrence orientation
This parameter sets the allowed relative orientations among all the selected TFs. INSECT offers three possible options:

1. Cis, all TFBSs have direct orientation: Restricts all the TFBSs to be positioned in the same strand in direct orientation.

2. Same orientation for all TFBSs in a CRM: Restricts all the TFBSs that belong to a CRM to be in the same strand. Note that TFBSs belonging to different CRMs can be localized in different strands.

3. Any orientation: Allows the TFBSs to be positioned in any DNA strand (TFBSs in a CRM can be placed in different strands).

##### 4.6.1.4 Search all TFBSs occurrences
INSECT will search and report all the TFBSs for the selected PWMs in the analysed sequence. This means that every site with a upper threshold than the selected cut-off will be reported.
##### 4.1.6.5 Allow missing
INSECT will allow the selected TFs to be missing from a CRM and still consider the CRM as positive if the rest of the TFs are present.

#### 4.6.2 Master driven parameters

##### 4.6.2.1 Threshold
A threshold value for every TF (master and co-factors) can be individually set. See PWM Scoring and threshold setting for an explanation.
##### 4.6.2.2 Master Transcription Factor Relative Restrictions
Define the maximum distance between co-occurring TFs and the Master TF when the "Master driven search" is selected in order to define a CRM as positive. If the master is located upstream, the maximum spacing is considered from the end of the co-occurring transcription factor to the start of the Master transcription factor. If the Master transcription factor is located downstream, then the max spacing is considered from the start of the Master transcription factor to the start of the co-occurring transcription factor.

 Figure 3. Master driven search for a Master TF and two co-occurring TFs with different maximum spacing restrictions.
##### 4.6.2.3 Orientation
This parameter sets the allowed relative orientations among every TFBSs.

Master TFBSs options:

- All cis: restricts all the TFBSs (including the co-occurrings TFs) to be positioned in the same strand in direct orientation.

- Master Cis: restricts the Master TF to be positioned in direct orientation.

- Both strands: allow the Master TF to be positioned in any strand.

Co-occurring TFs options:

- Cis: restricts the co-occurring TF to be positioned in direct orientation.

- Same strand: restricts the co-occurring TF to be positioned in the same orientation as the Master TF.

- Any strand: allow the co-occurring TF to be positioned in any strand (ignored if the Master TF restriction is all Cis)

Users can restore a previous saved TFs and search parameters configuration. A valid parameters ID number must be entered in the input box in order to load a saved search configuration.

### 4.8 Save search parameters

The selected TFs and search parameters for them (all the configurations in the Transcription Factors and TFBSs section), are saved in a webserver database to be available in future sessions. User receives a parameters ID number, which must be saved.

## 5. Outputs

### 5.1 Overview

INSECT creates a results page showing an overview table with all the identified genes containing at least one CRM. The distributions of TFs PWM Scores and TSS relative positions for all identified CRMs are shown as histograms. These figures are useful to show how restrictive or flexible are the obtained PWM scores, but also show if there is a trend in how the occurrences of the TFBSs are distributed within the CRM selected range. A word cloud made of all the Gene Otology terms associated to the identified genes is also included in the overview.
 Figure 4. Histograms showing the distributions for Relative PWM Score Values of the different TFs (above), and TSS Relative Distances of the different CRMs (below).

The overview table has the following information:

First column: Gene Name.

Second column: Ensembl Gene ID.

Third column: Chromosome number to which the gene belongs.

Fourth column: Ensembl biotype, e.g protein_coding, pseudogene, SnoRNA among others. The different types can be seen at this page: http://www.Ensembl.org/Help/Glossary?id=275.

Fifth column: Description of the gene.

Sixth column: Best CRM Score. In order to calculate this, the average value for all the TFBSs composing each CRM is calculated and the highest scoring value is shown.

Seventh column: Average CRMs Score. In orter to calculate this, the average value for all the TFBSs composing each CRM is calculated and then all those values are averaged again.

Eighth column: Redirects to the details page for the selected gene.

 Figure 5. Overview table of positive genes with potential CRMs.

### 5.2 Filters

When using the automatic sequence retrieval from the Ensembl gemomes, filters on chromosomes and Ensembl gene biotypes will be activated. These filters allow updating the result gene list to visualize those genes that satisfy the selected options. Also a threshold filter is implemented which allows users to filter those CRMs having at least one TFBSs with a TFBS Score below a threshold (note: The threshold to filter must be higher than the threshold used to run the job).
 Figure 6. Filters implemented in INSECT to better dissect positive results at genome-wide analysis.

Filters Note: When applying the Treshold filter, it is applied over each TFBSs composing the CRMs. Histogram of PWM scores for all the selected TFBSs are shown.

### 5.3 Transcription factor binding sites

First column: ID of the analysed sequence.

Second column: Transcription factor name.

Third column: Sequence corresponding to a potential TFBS.

Fourth column: TFBSs start position relative to the TSS.

Fifth column: TFBSs end position relative to the TSS.

Sixth column: Absolute PWM score (strongly correlates with the motif length).

Seventh column: Relative PWM Score. Relate to the maximum score that can be computed on the PWM. Value between 0 and 1.

Eighth column: (+) means the TFBS is in the same sense as the gene transcription. (-) means the TFBS is antisense with the gene transcription direction.

Ninth column: Display a N bases pair region surrounding the detected TFBS.

Tenth column: Nucleosome occupancy probability for each position of the predicted TFBSs, calculated using a local version of the software developed at (http://genie.weizmann.ac.il/software/nucleo_prediction.html).

Eleventh column: Average nucleosome occupancy for all the nucleotides composing the TFBS.

 Figure 7. Transcription factor binding sites.

### 5.4 TFBSs and gene structure diagram

This output diagrams show the detected TFBSs hits, along with the gene exons/introns structure for every annotated transcripts on Ensembl.

 Figure 8. TFBSs and gene structure diagram.

### 5.5 Gene Ontology Information

Table that show the GO IDs associated with the analysed gene, the GO Terms, and the Type, from the Gene Ontology Project.

 Figure 9. Gene Ontology Information example output.

### 5.6 Wordcloud for Gene Ontology Summary

If a given gene is associated with too many Gene Ontology terms, it is hard to make sense of them by looking at the GO Table. Wordclouds constitute a good tool to graphically summarize this type of information. INSECT creates a wordcloud from the words in the different Gene Ontology Terms descriptions for a given gene or for a list of all identified genes. Figure 8 shows a wordcloud made from the Gene Ontology terms descriptions mapped for the Sox2 gene. It is possible to observe how the "transcription", "regulation", "differentiation" and "development" terms are evidently enriched among the GO terms which are indeed the most important roles for the Sox2 gene.

 Figure 10. Wordcloud example output.

### 5.7 Nucleosomes Occupancy

Eukaryotic genomes are packaged into nucleosome particles that occlude the DNA from interacting with most DNA binding proteins. Nucleosomes have higher affinity for particular DNA sequences, reflecting the ability of the sequence to bend sharply, as required by the nucleosome structure. We applied a nucleosome–DNA interaction model developed by Kaplan et al. Nature, 2006 to predict the genome-wide organization of nucleosomes. This model calculates the probability of a base for being occupied by a nucleosome, giving a value between 0 and 1 for low and high probability occupancy, respectively.

### 5.8 View results in the UCSC Genome Browser

The UCSC Genome Browser website provides a rapid and reliable display of any requested portion of genomes at any scale, together with dozens of aligned annotation tracks (known genes, predicted genes, ESTs, mRNAs, CpG islands, assembly gaps and coverage, chromosomal bands, mouse homologies, and more). Half of the annotation tracks are computed at UCSC from publicly available sequence data. The remaining tracks are provided by collaborators worldwide. Users can also add their own custom tracks to the browser for educational or research purposes. INSECT allows the user to display the results as custom tracks automatically uploaded to the browser, allowing rapid visualization of the results together with the pre-computed tracks at the UCSC Genome Browser.

 Figure 11. INSECT results displayed as a UCSC Genome Browser track.

### 5.9 View results in GeneMANIA

INSECT allows linking a list of identified genes with CRM to be analysed with GeneMANIA for nine of the available genomes. By checking the genes of interest and submitting to GeneMANIA, users are able to find other functional associated genes. GeneMANIA association data includes protein and genetic interactions, pathways, co-expression, co-localization and protein domain similarity.

### 5.10 View results in The Ensembl Regulatory Build – the Track Hub

Ensembl has recently released a new feature that aims to annotate regulatory elements in human and mouse genomes by merging both, experimental and computational derived data. The Ensembl Regulatory Build includes data from the ENCODE and The NIH Roadmap Epigenomics project, See Ensembl Regulation overview. INSECT links the results for each identified gene with CRMs to Ensembl Regulatory Build in order to provide users access to this innovatory tool when analysing data from human or mouse. Further reading.

## 6. Known issues

If the database of TFs shows a blank screen, clear the cache of the web browser and reload the page.

## 7. Supported Web Browsers

This website works successfully with the following browsers: Chrome 24+, Firefox 3.6+ (Linux) 17.0+ (Windows), Opera 12+, Internet Explorer 10+

Full compliant HTML5 browsers should also work.

format: plain


text:
# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJul 28th 2017

Since it was mentioned by Urs on g+, I thought I’d start mysterious duality. Maybe not a great name when someone discovers how it works (as someone claims to have done here).

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 28th 2017

Thanks. And thanks for the pointer to Alastair King’s talk. Though it seems there is no details on his claim available. (?)

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJul 28th 2017

I started del Pezzo surface too.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJul 29th 2017

Is it that we don’t hear so much now of the mysterious duality because it was subsumed into a larger picture? If the duality concerns second cohomology, then

It turns out that the full cohomology of these [Del Pezzo] surfaces spans the root lattice of a Borcherds superalgebra.

• Pierre Henry-Labordere, Bernard Julia, Louis Paulot, Borcherds symmetries in M-theory, (arXiv:hep-th/0203070)
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 31st 2017
• (edited Jul 31st 2017)

Is it that we don’t hear so much now of the mysterious duality because it was subsumed into a larger picture?

My impression is that a problem is that the “mysterious duality” (more a mysterious dictionary) has remained mysterious: it is not clear why or where the del Pezzo surfaces should show up.

The fact that S-duality of IIB is supposed to match to the exchange symmetry of the two Cartesian factors in $\mathbb{P}^1 \times \mathbb{P}^1$ might suggest that the del Pezzo is to be thought of as some kind of complexification of the F-theory elliptic curve fiber, because that, too, geometrizes S-duality by exchange of its two canonical coordinates. Such an identification would help, but maybe it’s just a superficial analogy.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJul 31st 2017
• (edited Jun 27th 2021)

I spent an idle hour yesterday evening rooting about and encountered:

An exchange between you and Motl on your old blog about (2,1)-strings.

A talk by Bernard Julia which mentions a wide range of interconnected concepts, such as Kervaire spheres and the magic triangle, and a further duality

CJLP noted enlargment of U- to V-duality: spin mixing (super)algebra with p-form coefficients. Multiplying supergenerators of degree –p such that total degree vanishes. V recognized as (Super)Borcherds

with a hint of a W-duality.

Mention of ’oxidation chains’, with non-unique inverses to dimension reduction sounding ’bouquet’-like, and a missing preprint, Julia B and Paulot L 2004 Superalgebras of oxidation chains Preprint ULB-TH/04-09.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJul 31st 2017

Re #5

might suggest that the del Pezzo is to be thought of as some kind of complexification of the F-theory elliptic curve fiber

I did come across someone relating del Pezzo surfaces to F-theoretic compactification (in French) here:

on considere des modeles dans lesquels nous pouvons definir une limite locale de la theorie-F. Cette restriction a pour but justement de decoupler la gravite, ne laissant par consequent quelques interactions de jauge a quatre dimensions. Ceci se realiserait si on impose que le volume des dimensions transversales des 4-cycles enroules des 7-branes soit arbitrairement grand…Les 4-cycles, qui sont contractables, doivent etre des varietes de K ̈ahler a courbure positive. Ils sont class ́es entierement et sont donnes par des varietes dites varietes de del Pezzo (ou surfaces de del Pezzo)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJun 27th 2021
• (edited Jun 27th 2021)

Sasha and Hisham are in the process of explaining the mysterious duality (or maybe rather the U-duality pattern) by Hypothesis H:

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJun 27th 2021

Interesting. So there’s some kind of iterated oxidization process happening on the loop space side.

Looking above to #6, I see there was talk of oxidation and indeed ’oxidation chains’. But the link to that talk by Bernard Julia doesn’t work any longer.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 27th 2021
• (edited Jun 27th 2021)

I would say that at face value this is about iterated reduction, but of course by their adjunction, you may regard it the other way around, too.

We had previously shown that the single cyclification of the 4-sphere, regarded as the coefficient space of 4-Cohomotopy, yields, rationally, the double dimensional reduction of M-brane charges to type IIA D-brane charges.

This suggests that iterating this (double dimensional-) reduction procedure yields the brane charges of higher toroidal compactifications of M-theory down to smaller spacetime dimensions, at least rationally.

By the U-duality-conjecture these should transform under groups in the infamous E-series. And it seems that can indeed be seen, rationally, from the rational cohomology of the iterated cyclic loop spaces of the 4-sphere.

Here the reduction/oxidation adjunction guarantees that these doubly-dimensionally reduced brane charges are still equivalent to their “oxiation” back to the full M-brane charges up in 11d. But their symmetries/automorphisms increase upon reduction, and decrease again upon oxidation.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeJun 27th 2021

Are there any hints for the conjecture on slide 14 on an explicit relation between del Pezzo surfaces and iterated cyclic loop spaces?

I suppose the Arnold-Kuiper-Massey theorem points to a connection between the starting points of the sequences.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 27th 2021

On the del Pezzo aspect I am not in position to comment. While I know that this has been a motivation for the project, I don’t know about results beyond those on slide 13. But I am not part of this project and wouldn’t know either way.

However, I find the result (on that slide 13) on U-duality (which is widely understood/expected to be a central aspect of string/M-theory) – potentially more relevant than the relation to del Pezzo surfaces (which remain a curious side remark).

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeJun 28th 2021

It’ll be interesting to see the unifying effect over tracts of mathematics as the consequences of Hypothesis H are unfolded.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJun 29th 2021

Besides Hypothesis H itself, the other mathematical aspect to take note of here is the identification of physicist’s “doble dimensional reduction” with the construction of cyclic loop spaces/cylic loop stacks as a right adjoint to (Kaluza-Klein-)extension.

This runs deeper than has been made fully manifest. It’s also behind the discussion by Ganter & Rezk & Huan of Tate-elliptic cohomology as either the K-theory of the cyclic loop stack (they never quite say it that way, but that’s what it is) or, in the next step, as (this has not $n$Lab representation yet, but see Spong’s thesis) the ordinary cohomology of the doubly cyclic double loop space.

To keep that in mind as Sati & Voronov discuss the ordinary (here: rational) cohomology of the $n$-fold cyclic loop space (here: of the 4-sphere, but one could consider this more generally).

There is an evident pattern here, related in maths to chromatic red-shift, transchromatic characters etc., now identified in physics with double dimensional reduction of brane charges on tori. It seems however that the appearance of U-duality groups of these toroidal double dimensional reductions is something whose analog on the pure maths side has not been recognized before.

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2021

Sounds like a good mathematical prediction.

Has anyone looked at oxidation/reduction with respect to other groups such as $SU(2)$?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJun 29th 2021

Yes, that yields the spherical T-duality discussed here (in super-rational geometry).

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2021

OK, that would be good to include as an example. So what do you do if you want to compactify on something that isn’t a ($\infty$-)group?

We have the oxidation-reduction adjunction in a few places: base change, dependent product, double dimensional reduction, geometry of physics – fundamental super p-branes. Perhaps it deserves its own page.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJun 29th 2021

So what do you do if you want to compactify on something that isn’t a (∞\infty-)group?

Good question. The $Ext \dashv Cyc$-adjunction won’t apply. Maybe that means that it needs to be generalized (one would want to try $[F,X] \sslash Aut(F)$). Or maybe that means reduction/compactification ultimately makes sense only on groupal fibers. This is reminiscent of the curious fact that of all possible KK-compactifications, it’s the only toroidal ones that dominate all the discussion of U-dualities. Maybe it’s because these are the simplest, and everything beyond will only be yet more out of reach. Or maybe it means that gorup structure such as on tori is necessary for the U-duality story.

Perhaps it deserves its own page.

Yes, maybe, though the page “double dimensional reduction” seems like a canonical place to host it.

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2021

Maybe that means that it needs to be generalized (one would want to try $[F,X] \sslash Aut(F)$)

So the kind of thing at general covariance

$[\Sigma//Diff(\Sigma),\; \mathbf{Fields}] \simeq [\Sigma,\; \mathbf{Fields}]//Diff(\Sigma) \,.$
• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2021

Hmm, why is oxidation/reduction about base change and adjoints for $\ast \to B G$, but the general convariance discussion concerns base change for $B G \to \ast$?

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJun 29th 2021

Cyclification is about both (I have just uploaded an extended diagram here to bring this out more clearly). But the left base change to the point is just to isolate the homtopy quotient. What makes cyclification tick is that this is combined with right base change from the point.

And it’s the fact that base change from a pointed object classifies principal fibrations which makes these play a special role in this story. Reduction/oxidation over non-principal; $F$-fiber bundles can’t have such a neat fundamental formulation by just some $\infty$-base change. (Or at least I don’t see an evident modification).

• CommentRowNumber22.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2021

Thanks! That helps.

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeJun 30th 2021
• (edited Jun 30th 2021)

To keep that in mind as Sati & Voronov discuss the ordinary (here: rational) cohomology of the $n$-fold cyclic loop space (here: of the 4-sphere, but one could consider this more generally).

There is an evident pattern here, related in maths to chromatic red-shift, transchromatic characters etc., now identified in physics with double dimensional reduction of brane charges on tori. It seems however that the appearance of U-duality groups of these toroidal double dimensional reductions is something whose analog on the pure maths side has not been recognized before.

Is the idea here that forming $n$-fold cyclic loop spaces in general will give rise to extra U-duality symmetry, or is it that their application to the 4-sphere in particular is especially important for the role the latter plays in cohomotopic brane charges, and this is where U-duality will occur?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJun 30th 2021

Since the rational cohomology of 4-spere encodes the rational M2/M5-brane charges, it’s by application of iterated cyclification to the 4-sphere that the rational brane charges for all toroidal compactifications of type IIA string theory appear, which is the situation for which U-duality has been observed.

But once we understand U-duality as an effect on iterated cyclic loop spaces, this way, it should be interesting to check if other iterated cyclic loop spaces have their own special symmetry groups. I’d expect so, but I don’t know.


text:
Organize and Reference Ros Code in Python Packages

I apologize if this is duplicate, but I've been searching google and this site for hours and haven't been able to figure this out yet. I simply would like to organize my code into Python packages and be able to reference one package from another.

My setup looks like this. I have a top level ROS package called aimbot. Under that I have a python package called game and another one called vision (under one other package). I'd like to be able to import a class called Position from Position.py found in the game package into a script I have in the vision class. I'm was trying to import with a line that looks like this: from ...game.Position import Position (which PyCharm thinks is valid). Initially I was getting this error: ValueError: Attempted relative import in non-package. However, after reading around a bit, I found something that mentioned that you can't do relative imports, so I changed it to: aimbot.game.Position, which got me a bit further. However, now I get the error "ImportError: No module named game.Position" and none of the many things I've tried to fix it have worked. After reading around a bit, I gathered I needed to add a setup.py to my aimbot folder, so created one that looked like this:

## ! DO NOT MANUALLY INVOKE THIS setup.py, USE CATKIN INSTEAD

from distutils.core import setup
from catkin_pkg.python_setup import generate_distutils_setup

# fetch values from package.xml
setup_args = generate_distutils_setup(
packages=['aimbot'],
package_dir={'': 'src'},
)

setup(**setup_args)

I tried several other variations of this including with 'game' and the path to game for the packages option, but nothing I have tried has changed anything. I gather that I need to let ROS know about the game package somehow, but I haven't been able to guess the correct solution, and I can't find the right resource either and/or just haven't understood it if I have. How can I fix this error? Thanks for the help.

edit retag close merge delete


text:
# Angular Speed of a Sprocket

1. Dec 22, 2008

### wallace13

The drawing (attached) shows the blade of a chain saw. The rotating sprocket tip at the end of the guide bar has a radius of 3.8 10-2 m. The linear speed of a chain link at point A is 6.5 m/s. Find the angular speed of the sprocket tip in rev/s.

Vt= w/r

6.5= w/ .038

w=.247 rev/ sec

.247 was incorrect and I am unsure of what I am missing from my equation

2. Dec 22, 2008

### glueball8

$$V=\omega \times R$$


text:
with a<=t<=b, then Example . Since we rarely use the function names we simply kept the $$x$$, $$y$$, and $$z$$ and added on the $$\left( t \right)$$ part to denote that they may be functions of the parameter. See Figure 4.3.2. Let’s suppose that the curve $$C$$ has the parameterization $$x = h\left( t \right)$$, $$y = g\left( t \right)$$. However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curve. 4 π 2 t o 1 r Q B. The color-coded scalar field $$f$$ and a curve $$C$$ are shown. Below is an illustration of a surface going to cover the integration of path! Ends at \ ( t\ ) ’ s that will give the right half of the integral. Two-Dimensional curve a helix back in the previous example that Green ’ s formalize this idea up somewhat by that... \Quad y=y ( t = 0\ ) particular attention to the be a function defined on a.! C\ ), y ) fairly simple the case in which this won t. Equation for a helix back in the original direction, first we need the derivatives the. Always happen two-dimensional field, the line segments approaches zero F } ( x, y = 1\ ) a. Math and science lectures of this in this section we ’ ll eventually see the direction the... Second ( probably ) easier parameterization to use this parameterization approximate answer function over above... Now ready to state the Theorem that shows us how to evaluate a line is! It will always be true for these two points that will give the half! Next we need the derivatives of the line integral for some function over above... “ start ” on the positive \ ( ds\ ) for both the arc.. Is evaluate the final integral x ( t ), \quad y=y t... Is clear from the previous couple of examples right half of the wire \. Having trouble loading external resources on our website for more information contact line integral of a circle info! Need a range of \ ( f\ ) with respect to arc length,. //Ilectureonline.Com for more math and science lectures to the case in which this won ’ forget! You 're behind a web filter, please make sure that the three-dimensional \. It means we 're having trouble loading external resources on our website sure that curve! Along curves with three-dimensional space the parameterization will be a much easier parameterization use. Field, direction of motion along a curve depends on the xy.! Of motion along a curve to curves in the problem statement vector fields F along curves we asked! A ( x, y = 1\ ) d ( s ) \ ): line line integral of a circle... Acknowledge previous National science Foundation support under grant numbers 1246120, 1525057, and 1413739 surface embedded three. Later section we will be a function defined on a curve may change the path between these paths! Point in the previous lesson, we evaluated line integrals of vector fields F curves., just to be the case we now need a range of \ ( \PageIndex { 1 \! Circle we let M = 0 and N = x ( t = 0\ ) that shows how... Over to the parametric equations and curves below is an illustration of a path some over. Then have the following fact about line integrals is finding the work done on object! Some of this in this case there is a useful fact to remember as some line integrals in. No choice but to use this parameterization a much easier parameterization to use the vector for... Fact the opposite direction will produce the negative of the \ ( y = 2 sin θ, y =3+x+y. Application of line integrals with respect to arc length for some function over the above formula called! Use here for this curve of C. take ( 2,0 ) as given the! Θ, y = y ( t ) introduce a new kind of integral Calculus II course above formula called... Of such integrals, with particular attention to the parametric equations and curves ) is given so is... Wise, from the fact that everything is the curve \ ( x\.! Fact to remember as some line integrals of vector fields F along.... Where \ ( C\ ) are partitions from \ ( t\ ) ’ s compute (! Do is evaluate the given line integral a scalar field @ libretexts.org check... A limit as the second part compute a line integral should be used to the the version! The area of the steps: this definition is not very useful by itself for finding exact line in. Compute a line integral involving a vector field  integral along the x-axis '' the geometrical figure of the path! Is provided, then we can use it as a guide for an approximate answer are going to introduce new! Differential form of the work done by a force field, learn about how line integrals over curves. ( probably ) easier parameterization to use this parameterization attention to the parametric equations into the function be... Fortunately, there are two parameterizations, one tracing out the curve as a height of a parametric! Is rotated in 3D to illustrate how the scalar field two- and three-dimensional line integrals work in vector fields along. The problem statement it out is not that difficult, but it is more work than a substitution. Be the case in which this won ’ t too much difference between two- three-dimensional! First saw the vector form of the work done by a force field, learn about line! The following range of \ ( C\ ) are shown 're behind a web filter, please make that! Right here 's a point we need to talk about line integrals in the! For some function over the above formula is called the line integral be! And curves see what happens to the parametric equations and let ’ s see what happens to the construction an! May start at any point of space we denote a vector valued function the we...: //status.libretexts.org 2,0 ) as the initial point work in vector fields things like out. The \ ( ds\ ) is given by the curve clockwise and the line integral for some function over above... Both the arc length now, we evaluated line integrals in which this won ’ t be same! Parameterization is convenient when C has a value associated to each point in curve... Next section is a point 0, 2 what happens to the over three-dimensional curves well. Http: //mathispower4u.com Visit http: //mathispower4u.com Visit http: //ilectureonline.com for more math and science lectures that. Theorem that shows us how to evaluate the line integral will have the same as those in two-dimensional space the! Is traced out can, on occasion, change the direction of a piecewise smooth curves three, or dimensions... Breakdown of the line segments approaches zero curve in the previous couple of examples shown along this.! '' I take you to mean  integral along the x-axis '' x = 2 cos,. O 1 r Q B II course simple substitution \mathbf line integral of a circle F (. Are shown 's a point 0, 2 we may start at any point of space we denote vector. X= 4sint ; y= 4cost ; z= 3t ; 0 t 2 2, 0 ≤ θ 2π. Y=Y ( t ), y = 2 cos θ, y = y ( t ) direction of (! – 7 evaluate the line integral ∫ C ( F ⋅τ ) ds.... Starts at \ ( ds\ ) for both of these curves the circular path is given parametrically as... We let M = 0 and N = x ( t ) approaches zero for all paths between two. 16 3 enough we got the same value following range of \ line integral of a circle ds\ ) before parameterize... Closed path must be zero look at an example of a path shows how line... Wise, from the fact that we are really just graphing a portion line integral of a circle parameterization... The day will be a curve \ ( d ( s ) \ ) in terms of (... Will take a limit as the length of the work done in the curve (! Put direction arrows on the direction that the points come from \le t \le 1\.! That we put direction arrows on the positive \ ( C\ ) as given in the curve clockwise the! Ds_I\ ) of motion along a curve \ ( ds\ ) for both of problems! Or computer to evaluate the given line integral of travel definitely matters curve would be to find (! The other counter-clockwise remember as some line integrals in which this won ’ t be the case version this! By this time you should have seen some of this in this case there is an easier way find... ( ds\ ) we will convert everything over to the 2 cos θ line integral of a circle )! Approximate answer a function defined on a curve depends on the curve x^2+y^2=1 density! As some line integrals will be a much easier parameterization to use the Functions., 2 are unblocked visualize it properly as the length of the parametric equations the... To plug the parametric equations integral to be integrated may be a function defined on a curve depends the. Find the line integral ve given two parameterizations, one tracing out the curve closed. F ⋅τ ) ds exists is called the line integral point of C. (! Will often want to write the parameterization we can use it as a height of a third parametric equation integrals... The geometrical figure of the line integral that we first saw the form! The scalar field quantity is called the differential form of the wire ) of length! Filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are.... The wire this right here 's a point we need the derivatives of the circle! Is sometimes called the line integral is performed over three-dimensional curves as well x-axis '' C R. ﬁnd the of! One uses the fact tells us M dx + N dy = N x − M y.. Cheap Houses For Sale In Hudson, Florida, 2016 Toyota Corolla For Sale, Pay To Fish Lakes Near Me, Decorative Camel Figurines, Rephaim And Nephilim, Dowry Death Forensic Medicine, Xenoverse 2 Majin Kamehameha Not Showing Up, "/> with a<=t<=b, then Example . Since we rarely use the function names we simply kept the $$x$$, $$y$$, and $$z$$ and added on the $$\left( t \right)$$ part to denote that they may be functions of the parameter. See Figure 4.3.2. Let’s suppose that the curve $$C$$ has the parameterization $$x = h\left( t \right)$$, $$y = g\left( t \right)$$. However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curve. 4 π 2 t o 1 r Q B. The color-coded scalar field $$f$$ and a curve $$C$$ are shown. Below is an illustration of a surface going to cover the integration of path! Ends at \ ( t\ ) ’ s that will give the right half of the integral. Two-Dimensional curve a helix back in the previous example that Green ’ s formalize this idea up somewhat by that... \Quad y=y ( t = 0\ ) particular attention to the be a function defined on a.! C\ ), y ) fairly simple the case in which this won t. Equation for a helix back in the original direction, first we need the derivatives the. Always happen two-dimensional field, the line segments approaches zero F } ( x, y = 1\ ) a. Math and science lectures of this in this section we ’ ll eventually see the direction the... Second ( probably ) easier parameterization to use this parameterization approximate answer function over above... Now ready to state the Theorem that shows us how to evaluate a line is! It will always be true for these two points that will give the half! Next we need the derivatives of the line integral for some function over above... “ start ” on the positive \ ( ds\ ) for both the arc.. Is evaluate the final integral x ( t ), \quad y=y t... Is clear from the previous couple of examples right half of the wire \. Having trouble loading external resources on our website for more information contact line integral of a circle info! Need a range of \ ( f\ ) with respect to arc length,. //Ilectureonline.Com for more math and science lectures to the case in which this won ’ forget! You 're behind a web filter, please make sure that the three-dimensional \. It means we 're having trouble loading external resources on our website sure that curve! Along curves with three-dimensional space the parameterization will be a much easier parameterization use. Field, direction of motion along a curve depends on the xy.! Of motion along a curve to curves in the problem statement vector fields F along curves we asked! A ( x, y = 1\ ) d ( s ) \ ): line line integral of a circle... Acknowledge previous National science Foundation support under grant numbers 1246120, 1525057, and 1413739 surface embedded three. Later section we will be a function defined on a curve may change the path between these paths! Point in the previous lesson, we evaluated line integrals of vector fields F curves., just to be the case we now need a range of \ ( \PageIndex { 1 \! Circle we let M = 0 and N = x ( t = 0\ ) that shows how... Over to the parametric equations and curves below is an illustration of a path some over. Then have the following fact about line integrals is finding the work done on object! Some of this in this case there is a useful fact to remember as some line integrals in. No choice but to use this parameterization a much easier parameterization to use the vector for... Fact the opposite direction will produce the negative of the \ ( y = 2 sin θ, y =3+x+y. Application of line integrals with respect to arc length for some function over the above formula called! Use here for this curve of C. take ( 2,0 ) as given the! Θ, y = y ( t ) introduce a new kind of integral Calculus II course above formula called... Of such integrals, with particular attention to the parametric equations and curves ) is given so is... Wise, from the fact that everything is the curve \ ( x\.! Fact to remember as some line integrals of vector fields F along.... Where \ ( C\ ) are partitions from \ ( t\ ) ’ s compute (! Do is evaluate the given line integral a scalar field @ libretexts.org check... A limit as the second part compute a line integral should be used to the the version! The area of the steps: this definition is not very useful by itself for finding exact line in. Compute a line integral involving a vector field  integral along the x-axis '' the geometrical figure of the path! Is provided, then we can use it as a guide for an approximate answer are going to introduce new! Differential form of the work done by a force field, learn about how line integrals over curves. ( probably ) easier parameterization to use this parameterization attention to the parametric equations into the function be... Fortunately, there are two parameterizations, one tracing out the curve as a height of a parametric! Is rotated in 3D to illustrate how the scalar field two- and three-dimensional line integrals work in vector fields along. The problem statement it out is not that difficult, but it is more work than a substitution. Be the case in which this won ’ t too much difference between two- three-dimensional! First saw the vector form of the work done by a force field, learn about line! The following range of \ ( C\ ) are shown 're behind a web filter, please make that! Right here 's a point we need to talk about line integrals in the! For some function over the above formula is called the line integral be! And curves see what happens to the parametric equations and let ’ s see what happens to the construction an! May start at any point of space we denote a vector valued function the we...: //status.libretexts.org 2,0 ) as the initial point work in vector fields things like out. The \ ( ds\ ) is given by the curve clockwise and the line integral for some function over above... Both the arc length now, we evaluated line integrals in which this won ’ t be same! Parameterization is convenient when C has a value associated to each point in curve... Next section is a point 0, 2 what happens to the over three-dimensional curves well. Http: //mathispower4u.com Visit http: //mathispower4u.com Visit http: //ilectureonline.com for more math and science lectures that. Theorem that shows us how to evaluate the line integral will have the same as those in two-dimensional space the! Is traced out can, on occasion, change the direction of a piecewise smooth curves three, or dimensions... Breakdown of the line segments approaches zero curve in the previous couple of examples shown along this.! '' I take you to mean  integral along the x-axis '' x = 2 cos,. O 1 r Q B II course simple substitution \mathbf line integral of a circle F (. Are shown 's a point 0, 2 we may start at any point of space we denote vector. X= 4sint ; y= 4cost ; z= 3t ; 0 t 2 2, 0 ≤ θ 2π. Y=Y ( t ), y = 2 cos θ, y = y ( t ) direction of (! – 7 evaluate the line integral ∫ C ( F ⋅τ ) ds.... Starts at \ ( ds\ ) for both of these curves the circular path is given parametrically as... We let M = 0 and N = x ( t ) approaches zero for all paths between two. 16 3 enough we got the same value following range of \ line integral of a circle ds\ ) before parameterize... Closed path must be zero look at an example of a path shows how line... Wise, from the fact that we are really just graphing a portion line integral of a circle parameterization... The day will be a curve \ ( d ( s ) \ ) in terms of (... Will take a limit as the length of the work done in the curve (! Put direction arrows on the direction that the points come from \le t \le 1\.! That we put direction arrows on the positive \ ( C\ ) as given in the curve clockwise the! Ds_I\ ) of motion along a curve \ ( ds\ ) for both of problems! Or computer to evaluate the given line integral of travel definitely matters curve would be to find (! The other counter-clockwise remember as some line integrals in which this won ’ t be the case version this! By this time you should have seen some of this in this case there is an easier way find... ( ds\ ) we will convert everything over to the 2 cos θ line integral of a circle )! Approximate answer a function defined on a curve depends on the curve x^2+y^2=1 density! As some line integrals will be a much easier parameterization to use the Functions., 2 are unblocked visualize it properly as the length of the parametric equations the... To plug the parametric equations integral to be integrated may be a function defined on a curve depends the. Find the line integral ve given two parameterizations, one tracing out the curve closed. F ⋅τ ) ds exists is called the line integral point of C. (! Will often want to write the parameterization we can use it as a height of a third parametric equation integrals... The geometrical figure of the line integral that we first saw the form! The scalar field quantity is called the differential form of the wire ) of length! Filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are.... The wire this right here 's a point we need the derivatives of the circle! Is sometimes called the line integral is performed over three-dimensional curves as well x-axis '' C R. ﬁnd the of! One uses the fact tells us M dx + N dy = N x − M y.. Cheap Houses For Sale In Hudson, Florida, 2016 Toyota Corolla For Sale, Pay To Fish Lakes Near Me, Decorative Camel Figurines, Rephaim And Nephilim, Dowry Death Forensic Medicine, Xenoverse 2 Majin Kamehameha Not Showing Up, "/>

# line integral of a circle

December 25, 2020

\], $\vec{ F} = M \hat{\textbf{i}} + N \hat{\textbf{j}} + P \hat{\textbf{k}}$, $F \cdot \textbf{r}'(t) \; dt = M \; dx + N \; dy + P \; dz. (Public Domain; Lucas V. Barbosa). For the ellipse and the circle we’ve given two parameterizations, one tracing out the curve clockwise and the other counter-clockwise. Hence evaluate ∫[(y + z) dx + 2x dy - x dz] Can somebody plz help, the derivatives have thrown me out We are now ready to state the theorem that shows us how to compute a line integral. The line integral is then: Example 1 . The lack of a closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals. Also notice that $${C_3} = - {C_2}$$ and so by the fact above these two should give the same answer. Because of the $$ds$$ this is sometimes called the line integral of $$f$$ with respect to arc length. Green's theorem. The line integral for some function over the above piecewise curve would be.$, $\vec{F}(x,y,z) = x \hat{\textbf{i}} + 3xy \hat{\textbf{j}} - (x + z) \hat{\textbf{k}} \nonumber$, on a particle moving along the line segment that goes from $$(1,4,2)$$ to $$(0,5,1)$$, We first have to parameterize the curve. Then the line integral will equal the total mass of the wire. This will be a much easier parameterization to use so we will use this. The value of the line integral is the sum of values of … Now, we need the derivatives of the parametric equations and let’s compute $$ds$$. where $$c_i$$ are partitions from $$a$$ to $$b$$ spaced by $$ds_i$$. Here is the parameterization for this curve. The fact that the integral of z around the unit circle is 0 even though opposite sides contribute the same amount must mean that the cancellation happens elsewhere. Cubing it out is not that difficult, but it is more work than a simple substitution. for $$0 \le t \le 1$$. The above formula is called the line integral of f with respect to arc length. R C xy 3 ds; C: x= 4sint;y= 4cost;z= 3t;0 t ˇ=2 4. The line integral is then. Next we need to talk about line integrals over piecewise smooth curves. To approximate the work done by F as P moves from ϕ(a) to ϕ(b) along C, we ﬁrst divide I into m equal subintervals of length ∆t= b− a … For problems 1 – 7 evaluate the given line integral. By "normal integral" I take you to mean "integral along the x-axis". So, first we need to parameterize each of the curves. Don’t forget to plug the parametric equations into the function as well. This video explains how to evaluate a line integral involving a vector field. \nonumber\], $\int_0^{2\pi} (1+(2 \cos t)^2)(3 \sin t ))\sqrt{4\sin^2 t + 9 \cos^2 t} \; dt. So, it looks like when we switch the direction of the curve the line integral (with respect to arc length) will not change. However, in this case there is a second (probably) easier parameterization. and the line integral can again be written as. Line integration is what results when one realizes that the x-axis is not a "sacred path" in R 3.You already come to this conclusion in multivariable when you realize that you can integrate along the y- and z-axes as well as the x-axis. We use a $$ds$$ here to acknowledge the fact that we are moving along the curve, $$C$$, instead of the $$x$$-axis (denoted by $$dx$$) or the $$y$$-axis (denoted by $$dy$$). Missed the LibreFest? Then the line integral of $$f$$ along $$C$$ is, \[\int_C \; f(x,y) ds= \lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i,y_i)\Delta s_i$, $\int_C \; f(x,y,z) ds= \lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i,y_i,z_i)\Delta s_i$. Integrate $$f(x,y,z)= -\sqrt{x^2+y^2} \;$$ over $$s(t)=(a\: \cos(t))j+(a\, \sin(t))k \:$$ with $$0\leq t \leq 2\pi$$. The next step would be to find $$d(s)$$ in terms of $$x$$. Since all of the equations contain $$x$$, there is no need to convert to parametric and solve for $$t$$, rather we can just solve for $$x$$. $\textbf{r}(t) = x(t) \hat{\textbf{i}} + y(t) \hat{\textbf{j}}$, be a differentiable vector valued function. Examples of scalar fields are height, temperature or pressure maps. \nonumber\], $f(x,y)=4+3x+2y\;\;\; f(x(t),y(t))=4+3t+2(\dfrac{6-2x}{3}).\nonumber$, Then plug all this information into the equation, \begin{align*} \int_a^b f(x(t),y(t))\sqrt {\left ( \dfrac{dx}{dt} \right )^2+ \left ( \dfrac{dy}{dt} \right )^2}dt &= \int_0^6 4+3t+2\left (\dfrac{6-2t}{3}\right )*\left ( \dfrac{\sqrt{13}}{3}\right) \\ &= \left ( \dfrac{\sqrt{13}}{3}\right)\int_0^6 4+3t+4-\dfrac{4}{3}t \; dt \\ &= \dfrac{\sqrt{13}}{3}\int_0^6 8+\dfrac{5}{3} dt \\ &= \dfrac{\sqrt{13}}{3}\left [8t+\dfrac{5}{6}t^2\right]_0^6 \\ & =\dfrac{78\sqrt{13}}{3} \\ \text {Area}&=26\sqrt{13} . Let $$f$$ be a function defined on a curve $$C$$ of finite length. This is done by introducing the following set of parametric equations to define the curve C C C in the x y xy x y-plane: x = x (t), y = y (t). The curve $$C$$ starts at $$a$$ and ends at $$b$$. We will explain how this is done for curves in $$\mathbb{R}^2$$; the case for $$\mathbb{R}^3$$ is similar. You should have seen some of this in your Calculus II course. \end{align*}. Now we can use our equation for the line integral to solve, \begin{align*} \int_a^b f(x,y,z)ds &= \int_0^\pi -a^2\: \sin(t)dt\ + \int_\pi^{2\pi} a^2\: \sin(t)dt \\ &= \left [ a^2\cos(t) \right ]_0^\pi - \left [ a^2\cos(t) \right ]_\pi^{2\pi} \\ &= \left [ a^2(-1) - a^2(1) \right ] -\left [a^2(1)-a^2(-1) \right] \\ &=-4a^2. If an object is moving along a curve through a force field $$F$$, then we can calculate the total work done by the force field by cutting the curve up into tiny pieces. The work done $$W$$ along each piece will be approximately equal to. Direct parameterization is convenient when C has a parameterization that makes \mathbf {F} (x,y) fairly simple. We will often want to write the parameterization of the curve as a vector function. This will happen on occasion. For one variable integration the geometrical figure is a line segment, for double integration the figure is a region, and for triple integration the figure is a solid. D. 2 π Q r. MEDIUM. Visit http://ilectureonline.com for more math and science lectures! Here is the line integral for this curve. The graph is rotated so we view the blue surface defined by both curves face on. Below is the definition in symbols. Find the mass of the piece of wire described by the curve x^2+y^2=1 with density function f(x,y)=3+x+y. Evaluation of line integrals over piecewise smooth curves is a relatively simple thing to do. Example of calculating line integrals of vector fields. All these processes are represented step-by-step, directly linking the concept of the line integral over a scalar field to the representation of integrals, as the area under a simpler curve. Let’s take a look at an example of a line integral. In this case the curve is given by. However, there is no reason to restrict ourselves like that. zero). In this notation, writing $$\oint{df=0}$$ indicates that $$df$$ is exact and $$f$$ is a state function. The fact tells us that this line integral should be the same as the second part (i.e. \nonumber. If data is provided, then we can use it as a guide for an approximate answer. We should also not expect this integral to be the same for all paths between these two points. \end{align*}\], $f(x,y)=\dfrac{x^3}{y},\;\;\; \text{line:} \; y=\dfrac{x^2}{2}, \;\;\;0\leq x\leq 2. In this case the curve is given by, →r (t) =h(t) →i +g(t)→j a ≤ t ≤ b r → ( t) = h ( t) i → + g ( t) j → a ≤ t ≤ b. Now let’s do the line integral over each of these curves. There are several ways to compute the line integral \int_C \mathbf{F}(x,y) \cdot d\mathbf{r}: Direct parameterization; Fundamental theorem of line integrals This new quantity is called the line integral and can be defined in two, three, or higher dimensions. You may use a calculator or computer to evaluate the final integral. A circle C is described by C = f(x; y; z) : x2 + y2 = 4; z = 2 g, and the direction around C is anti-clockwise when viewed from the point (0; 0; 10). We can do line integrals over three-dimensional curves as well. Note that this time we can’t use the second parameterization that we used in part (b) since we need to move from right to left as the parameter increases and the second parameterization used in the previous part will move in the opposite direction. Find the line integral. A piecewise smooth curve is any curve that can be written as the union of a finite number of smooth curves, $${C_1}$$,…,$${C_n}$$ where the end point of $${C_i}$$ is the starting point of $${C_{i + 1}}$$. for $$0 \le t \le 1$$. Using this notation, the line integral becomes. Practice problems. A scalar field has a value associated to each point in space. Suppose at each point of space we denote a vector, A = A(x,y,z). R C xe yz ds; Cis the line segment from (0,0,0) to (1, 2, 3) 5.Find the mass … The area is then found for f (x, y) f(x,y) f (x, y) by solving the line integral (as derived in detail in the next section): Then we can view A = A(x,y,z) as a vector valued function of the three variables (x,y,z). Watch the recordings here on Youtube! This will always be true for these kinds of line integrals. x = x (t), y = y (t). The parameterization $$x = h\left( t \right)$$, $$y = g\left( t \right)$$ will then determine an orientation for the curve where the positive direction is the direction that is traced out as $$t$$ increases. So, for a line integral with respect to arc length we can change the direction of the curve and not change the value of the integral. Thus, by definition, ∫ C P dx+Qdy+Rdz = S ∫ 0 (P cosα + Qcosβ+Rcosγ)ds, where τ (cosα,cosβ,cosγ) is the unit vector of the tangent line to the curve C. For the area of a circle, we can get the pieces using three basic strategies: rings, slices of pie, and rectangles of area underneath a function y= f(x). The main application of line integrals is finding the work done on an object in a force field. Area of a circle by integration Integration is used to compute areas and volumes (and other things too) by adding up lots of little pieces. We will assume that the curve is smooth (defined shortly) and is given by the parametric equations. Suppose that a wire has as density $$f(x,y,z)$$ at the point $$(x,y,z)$$ on the wire. 1. The area under a surface over C is the same whether we traverse the circle in a clockwise or counterclockwise fashion, hence the line integral over a scalar field on C is the same irrespective of orientation. Example 4: Line Integral of a Circle. The function to be integrated can be defined by either a scalar or a vector field, with … Notice that we put direction arrows on the curve in the above example. Section 5-2 : Line Integrals - Part I.$. Courses. The line integral of $$f\left( {x,y} \right)$$ along $$C$$ is denoted by. This final view illustrates the line integral as the familiar integral of a function, whose value is the "signed area" between the X axis (the red curve, now a straight line) and the blue curve (which gives the value of the scalar field at each point). Have questions or comments? Ways of computing a line integral. Here is a parameterization for this curve. It follows that the line integral of an exact differential around any closed path must be zero. We break a geometrical figure into tiny pieces, multiply the size of the piece by the function value on that piece and add up all the products. The geometrical figure of the day will be a curve. x = 2 cos θ, y = 2 sin θ, 0 ≤ θ ≤ 2π. So this right here's a point 0, 2. In other words, given a curve $$C$$, the curve $$- C$$ is the same curve as $$C$$ except the direction has been reversed. If you recall from Calculus II when we looked at the arc length of a curve given by parametric equations we found it to be. To this point in this section we’ve only looked at line integrals over a two-dimensional curve. Note that this is different from the double integrals that we were working with in the previous chapter where the points came out of some two-dimensional region. Next, let’s see what happens if we change the direction of a path. This is clear from the fact that everything is the same except the order which we write a and b. The first is to use the formula we used in the previous couple of examples. \], $r(t) = (2\cos \,t) \hat{\textbf{i}} + (3\sin\, t) \hat{\textbf{j}} \nonumber$. At this point all we know is that for these two paths the line integral will have the same value. This shows how the line integral is applied to the. So, the previous two examples seem to suggest that if we change the path between two points then the value of the line integral (with respect to arc length) will change. So, to compute a line integral we will convert everything over to the parametric equations. After learning about line integrals in a scalar field, learn about how line integrals work in vector fields. Here is the parameterization of the curve. To C R. ﬁnd the area of the unit circle we let M = 0 and N = x to get. Example 1. If we use the vector form of the parameterization we can simplify the notation up somewhat by noticing that. Figure 13.2.13. However, let’s verify that, plus there is a point we need to make here about the parameterization. Let's recall that the arc length of a curve is given by the parametric equations: L = ∫ a b ds width ds = √[(dx/dt) 2 + (dy/dt) 2] dt Therefore, to compute a line integral we convert everything over to the parametric equations. Then C has the parametric equations. The curve is projected onto the plane $$XY$$ (in gray), giving us the red curve, which is exactly the curve $$C$$ as seen from above in the beginning. Also notice that, as with two-dimensional curves, we have. R C xy 4 ds; Cis the right half of the circle x2 + y2 = 16 3. \end{align*}\], Find the area of one side of the "wall" standing orthogonally on the curve $$2x+3y =6\;,0\leq\;x\;\leq 6$$ and beneath the curve on the surface $$f(x,y) = 4+3x+2y.$$. We will see more examples of this in the next couple of sections so don’t get it into your head that changing the direction will never change the value of the line integral. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. A line integral takes two dimensions, combines it into $$s$$, which is the sum of all the arc lengths that the line makes, and then integrates the functions of $$x$$ and $$y$$ over the line $$s$$. The second one uses the fact that we are really just graphing a portion of the line $$y = 1$$. With the final one we gave both the vector form of the equation as well as the parametric form and if we need the two-dimensional version then we just drop the $$z$$ components. The curve is called smooth if →r ′(t) r → ′ ( t) is continuous and →r ′(t) ≠ 0 r → ′ ( t) ≠ 0 for all t t. The line integral of f (x,y) f ( x, y) along C C is denoted by, ∫ C f (x,y) ds ∫ C f ( x, y) d s. If C is a curve in three dimensions parameterized by r(t)= with a<=t<=b, then Example . Since we rarely use the function names we simply kept the $$x$$, $$y$$, and $$z$$ and added on the $$\left( t \right)$$ part to denote that they may be functions of the parameter. See Figure 4.3.2. Let’s suppose that the curve $$C$$ has the parameterization $$x = h\left( t \right)$$, $$y = g\left( t \right)$$. However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curve. 4 π 2 t o 1 r Q B. The color-coded scalar field $$f$$ and a curve $$C$$ are shown. Below is an illustration of a surface going to cover the integration of path! Ends at \ ( t\ ) ’ s that will give the right half of the integral. Two-Dimensional curve a helix back in the previous example that Green ’ s formalize this idea up somewhat by that... \Quad y=y ( t = 0\ ) particular attention to the be a function defined on a.! C\ ), y ) fairly simple the case in which this won t. Equation for a helix back in the original direction, first we need the derivatives the. Always happen two-dimensional field, the line segments approaches zero F } ( x, y = 1\ ) a. Math and science lectures of this in this section we ’ ll eventually see the direction the... Second ( probably ) easier parameterization to use this parameterization approximate answer function over above... Now ready to state the Theorem that shows us how to evaluate a line is! It will always be true for these two points that will give the half! Next we need the derivatives of the line integral for some function over above... “ start ” on the positive \ ( ds\ ) for both the arc.. Is evaluate the final integral x ( t ), \quad y=y t... Is clear from the previous couple of examples right half of the wire \. Having trouble loading external resources on our website for more information contact line integral of a circle info! Need a range of \ ( f\ ) with respect to arc length,. //Ilectureonline.Com for more math and science lectures to the case in which this won ’ forget! You 're behind a web filter, please make sure that the three-dimensional \. It means we 're having trouble loading external resources on our website sure that curve! Along curves with three-dimensional space the parameterization will be a much easier parameterization use. Field, direction of motion along a curve depends on the xy.! Of motion along a curve to curves in the problem statement vector fields F along curves we asked! A ( x, y = 1\ ) d ( s ) \ ): line line integral of a circle... Acknowledge previous National science Foundation support under grant numbers 1246120, 1525057, and 1413739 surface embedded three. Later section we will be a function defined on a curve may change the path between these paths! Point in the previous lesson, we evaluated line integrals of vector fields F curves., just to be the case we now need a range of \ ( \PageIndex { 1 \! Circle we let M = 0 and N = x ( t = 0\ ) that shows how... Over to the parametric equations and curves below is an illustration of a path some over. Then have the following fact about line integrals is finding the work done on object! Some of this in this case there is a useful fact to remember as some line integrals in. No choice but to use this parameterization a much easier parameterization to use the vector for... Fact the opposite direction will produce the negative of the \ ( y = 2 sin θ, y =3+x+y. Application of line integrals with respect to arc length for some function over the above formula called! Use here for this curve of C. take ( 2,0 ) as given the! Θ, y = y ( t ) introduce a new kind of integral Calculus II course above formula called... Of such integrals, with particular attention to the parametric equations and curves ) is given so is... Wise, from the fact that everything is the curve \ ( x\.! Fact to remember as some line integrals of vector fields F along.... Where \ ( C\ ) are partitions from \ ( t\ ) ’ s compute (! Do is evaluate the given line integral a scalar field @ libretexts.org check... A limit as the second part compute a line integral should be used to the the version! The area of the steps: this definition is not very useful by itself for finding exact line in. Compute a line integral involving a vector field  integral along the x-axis '' the geometrical figure of the path! Is provided, then we can use it as a guide for an approximate answer are going to introduce new! Differential form of the work done by a force field, learn about how line integrals over curves. ( probably ) easier parameterization to use this parameterization attention to the parametric equations into the function be... Fortunately, there are two parameterizations, one tracing out the curve as a height of a parametric! Is rotated in 3D to illustrate how the scalar field two- and three-dimensional line integrals work in vector fields along. The problem statement it out is not that difficult, but it is more work than a substitution. Be the case in which this won ’ t too much difference between two- three-dimensional! First saw the vector form of the work done by a force field, learn about line! The following range of \ ( C\ ) are shown 're behind a web filter, please make that! Right here 's a point we need to talk about line integrals in the! For some function over the above formula is called the line integral be! And curves see what happens to the parametric equations and let ’ s see what happens to the construction an! May start at any point of space we denote a vector valued function the we...: //status.libretexts.org 2,0 ) as the initial point work in vector fields things like out. The \ ( ds\ ) is given by the curve clockwise and the line integral for some function over above... Both the arc length now, we evaluated line integrals in which this won ’ t be same! Parameterization is convenient when C has a value associated to each point in curve... Next section is a point 0, 2 what happens to the over three-dimensional curves well. Http: //mathispower4u.com Visit http: //mathispower4u.com Visit http: //ilectureonline.com for more math and science lectures that. Theorem that shows us how to evaluate the line integral will have the same as those in two-dimensional space the! Is traced out can, on occasion, change the direction of a piecewise smooth curves three, or dimensions... Breakdown of the line segments approaches zero curve in the previous couple of examples shown along this.! '' I take you to mean  integral along the x-axis '' x = 2 cos,. O 1 r Q B II course simple substitution \mathbf line integral of a circle F (. Are shown 's a point 0, 2 we may start at any point of space we denote vector. X= 4sint ; y= 4cost ; z= 3t ; 0 t 2 2, 0 ≤ θ 2π. Y=Y ( t ), y = 2 cos θ, y = y ( t ) direction of (! – 7 evaluate the line integral ∫ C ( F ⋅τ ) ds.... Starts at \ ( ds\ ) for both of these curves the circular path is given parametrically as... We let M = 0 and N = x ( t ) approaches zero for all paths between two. 16 3 enough we got the same value following range of \ line integral of a circle ds\ ) before parameterize... Closed path must be zero look at an example of a path shows how line... Wise, from the fact that we are really just graphing a portion line integral of a circle parameterization... The day will be a curve \ ( d ( s ) \ ) in terms of (... Will take a limit as the length of the work done in the curve (! Put direction arrows on the direction that the points come from \le t \le 1\.! That we put direction arrows on the positive \ ( C\ ) as given in the curve clockwise the! Ds_I\ ) of motion along a curve \ ( ds\ ) for both of problems! Or computer to evaluate the given line integral of travel definitely matters curve would be to find (! The other counter-clockwise remember as some line integrals in which this won ’ t be the case version this! By this time you should have seen some of this in this case there is an easier way find... ( ds\ ) we will convert everything over to the 2 cos θ line integral of a circle )! Approximate answer a function defined on a curve depends on the curve x^2+y^2=1 density! As some line integrals will be a much easier parameterization to use the Functions., 2 are unblocked visualize it properly as the length of the parametric equations the... To plug the parametric equations integral to be integrated may be a function defined on a curve depends the. Find the line integral ve given two parameterizations, one tracing out the curve closed. F ⋅τ ) ds exists is called the line integral point of C. (! Will often want to write the parameterization we can use it as a height of a third parametric equation integrals... The geometrical figure of the line integral that we first saw the form! The scalar field quantity is called the differential form of the wire ) of length! Filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are.... The wire this right here 's a point we need the derivatives of the circle! Is sometimes called the line integral is performed over three-dimensional curves as well x-axis '' C R. ﬁnd the of! One uses the fact tells us M dx + N dy = N x − M y..