Difference between revisions of "Vector analysis"
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Divergence of a vector field is the density of outward [[Flux|flux]] at a given point in <math>\mathbb{R}^3</math>. Under this definition, divergence is often thought of as 'flux density' which is the motivation for the [[Divergence theorem]]. | Divergence of a vector field is the density of outward [[Flux|flux]] at a given point in <math>\mathbb{R}^3</math>. Under this definition, divergence is often thought of as 'flux density' which is the motivation for the [[Divergence theorem]]. | ||
− | '''Definition''': Let <math>\mathbf{F}:U\to\mathbb{R}^3</math> where <math>U</math> is a subset of <math>\mathbb{R}^3</math>. Then the divergence of <math>\mathbf{F}</math> evaluated at the point <math>p\in U</math> is given by <cmath>\text{div}(\mathbf{F})(p) = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}\mathbf{F}\cdot\mathbf{n}\,dS,</cmath> where <math>R</math> is a region whose volume <math>V(R)</math> shrinks to <math>0</math> about the point <math>p</math>, <math>\partial R</math> is the boundary of <math>R</math>, <math>\mathbf{n}</math> is the outward unit normal of <math>\mathbf{F}</math> relative to <math>R</math>, and <math>dS</math> is an area element of <math>R</math>. | + | '''Definition''': Let <math>\mathbf{F}:U\to\mathbb{R}^3</math> where <math>U</math> is a subset of <math>\mathbb{R}^3</math>. Then the ''divergence'' of <math>\mathbf{F}</math> evaluated at the point <math>p\in U</math> is given by <cmath>\text{div}(\mathbf{F})(p) = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}\mathbf{F}\cdot\mathbf{n}\,dS,</cmath> where <math>R</math> is a region whose volume <math>V(R)</math> shrinks to <math>0</math> about the point <math>p</math>, <math>\partial R</math> is the boundary of <math>R</math>, <math>\mathbf{n}</math> is the outward unit normal of <math>\mathbf{F}</math> relative to <math>R</math>, and <math>dS</math> is an area element of <math>R</math>. |
==== Curl ==== | ==== Curl ==== | ||
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An alternative definition that one can give that does not rely upon the normal component of curl is as follows: | An alternative definition that one can give that does not rely upon the normal component of curl is as follows: | ||
− | '''Definition''': Let <math>\mathbf{F}:U\to\mathbb{R}^3</math> where <math>U</math> is a subset of <math>\mathbb{R}^3</math>. Then the curl of <math>\mathbf{F}</math> evaluated at the point <math>p\in U</math> is given by <cmath>\text{curl}(\mathbf{F})(p) = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}\mathbf{F}\times\mathbf{n}\,dS,</cmath> where <math>R</math> is a region whose volume <math>V(R)</math> shrinks to <math>0</math> about the point <math>p</math>, <math>\partial R</math> is the boundary of <math>R</math>, <math>\mathbf{n}</math> is the outward unit normal of <math>\mathbf{F}</math> relative to <math>R</math>, <math>\mathbf{t}</math> is the unit tangent of <math>\mathbf{F}</math> relative to <math>R</math>, and <math>dS</math> is an area element of <math>R</math>. | + | '''Definition''': Let <math>\mathbf{F}:U\to\mathbb{R}^3</math> where <math>U</math> is a subset of <math>\mathbb{R}^3</math>. Then the ''curl'' of <math>\mathbf{F}</math> evaluated at the point <math>p\in U</math> is given by <cmath>\text{curl}(\mathbf{F})(p) = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}\mathbf{F}\times\mathbf{n}\,dS,</cmath> where <math>R</math> is a region whose volume <math>V(R)</math> shrinks to <math>0</math> about the point <math>p</math>, <math>\partial R</math> is the boundary of <math>R</math>, <math>\mathbf{n}</math> is the outward unit normal of <math>\mathbf{F}</math> relative to <math>R</math>, <math>\mathbf{t}</math> is the unit tangent of <math>\mathbf{F}</math> relative to <math>R</math>, and <math>dS</math> is an area element of <math>R</math>. |
==== Gradient ==== | ==== Gradient ==== | ||
The gradient of a scalar field <math>f:\mathbb{R}^3\to\mathbb{R}</math> is typically interpreted to be the operator that gives the vector whose direction is that in the direction of greatest increase of <math>f</math> and magnitude being the rate at which <math>f</math> is increasing. As such, we can define the gradient in a similar way to divergence and curls as below: | The gradient of a scalar field <math>f:\mathbb{R}^3\to\mathbb{R}</math> is typically interpreted to be the operator that gives the vector whose direction is that in the direction of greatest increase of <math>f</math> and magnitude being the rate at which <math>f</math> is increasing. As such, we can define the gradient in a similar way to divergence and curls as below: | ||
− | '''Definition''': Let <math>f:U\to\mathbb{R}</math> where <math>U</math> is a subset of <math>\mathbb{R}^3</math>. Then the gradient of <math>f</math> evaluated at the point <math>p\in U</math> is given by <cmath>\text{grad}(f)(p) = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}f\mathbf{n}\,dS,</cmath> where <math>R</math> is a region whose volume <math>V(R)</math> shrinks to <math>0</math> about the point <math>p</math>, <math>\partial R</math> is the boundary of <math>R</math>, <math>\mathbf{n}</math> is the outward unit normal relative to <math>R</math>, and <math>dS</math> is an area element of <math>R</math>. | + | '''Definition''': Let <math>f:U\to\mathbb{R}</math> where <math>U</math> is a subset of <math>\mathbb{R}^3</math>. Then the ''gradient'' of <math>f</math> evaluated at the point <math>p\in U</math> is given by <cmath>\text{grad}(f)(p) = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}f\mathbf{n}\,dS,</cmath> where <math>R</math> is a region whose volume <math>V(R)</math> shrinks to <math>0</math> about the point <math>p</math>, <math>\partial R</math> is the boundary of <math>R</math>, <math>\mathbf{n}</math> is the outward unit normal relative to <math>R</math>, and <math>dS</math> is an area element of <math>R</math>. |
+ | |||
+ | ==== Laplacian ==== | ||
+ | The Laplacian of a scalar field <math>f:\mathbb{R}^3\to\mathbb{R}</math> is typically defined as the divergence of the gradient of <math>f</math> as below: | ||
+ | |||
+ | '''Definition''': Let <math>f:\mathbb{R}^3\to\mathbb{R}</math> be a scalar field. Then the ''Laplacian'' of <math>f</math> is given by <math>div(grad(f)).</math> | ||
{{stub}}[[Category:Calculus]] | {{stub}}[[Category:Calculus]] |
Revision as of 04:56, 21 December 2022
Vector analysis or vector calculus is the mathematical field dedicated to studying the methods of calculus such as differentiation and integration applied to vector fields. In modern mathematics, vector analysis is often taken to be sub-field of differential geometry. In terms of university course listings, it is common to use the word "vector calculus" synonymously with multivariable calculus.
Application-wise, vector analysis plays significant roles in the sciences and engineering. In physics, vector analysis is used heavily in the study of electromagnetism among various other fields of physics. In engineering, vector analysis often shows up in the form of the Cauchy stress tensor. Another significant application of vector analysis is to fluid dynamics. In particular, one can describe the Euler equations via the methods of vector analysis.
Contents
[hide]Classical Vector Analysis
Classical vector analysis largely is based developing the methods of calculus for . At the time when vector analysis was relatively new, it was common to perceive applied mathematics from only a 3-dimensional perspective as it is commonly propagated that the physical space that is observable is that of 3-dimensional Euclidean space. This is not completely true and necessitates generalizations of traditional vector analysis.
Scalar fields
A scalar field is traditionally a map where is a subset of . In other words, is a map that assigns a real-valued scalar to every point in . Generally, this scalar will represent some type of quantity such as potential in physics.
Vector fields
A vector field in the traditional sense is a map where is a subset of . That is, associates a vector in to every point in . Typically, this construction is used to represent some kind of direction along with a quantity being associated with to a specific point in . This may be electric fields, magnetic fields, or even fields that model fluid flow.
Differential operators
In the world of vector analysis, various forms of differential operators exist. Most notable are that of divergence, curl, the gradient, and the Laplacian.
Divergence
Divergence of a vector field is the density of outward flux at a given point in . Under this definition, divergence is often thought of as 'flux density' which is the motivation for the Divergence theorem.
Definition: Let where is a subset of . Then the divergence of evaluated at the point is given by where is a region whose volume shrinks to about the point , is the boundary of , is the outward unit normal of relative to , and is an area element of .
Curl
Curl of a vector field is the density of circulation at a given point in . Thus, one can think of curl intuitively as 'circulation density' which is motivation for Stokes' Theorem.
Definition: Let where is a subset of . Then the curl of evaluated at the point is given by where is a region whose area shrinks to about the point , is the boundary (closed loop) of , is the outward unit normal of relative to , is the unit tangent of relative to , and is an arclength element of .
An alternative definition that one can give that does not rely upon the normal component of curl is as follows:
Definition: Let where is a subset of . Then the curl of evaluated at the point is given by where is a region whose volume shrinks to about the point , is the boundary of , is the outward unit normal of relative to , is the unit tangent of relative to , and is an area element of .
Gradient
The gradient of a scalar field is typically interpreted to be the operator that gives the vector whose direction is that in the direction of greatest increase of and magnitude being the rate at which is increasing. As such, we can define the gradient in a similar way to divergence and curls as below:
Definition: Let where is a subset of . Then the gradient of evaluated at the point is given by where is a region whose volume shrinks to about the point , is the boundary of , is the outward unit normal relative to , and is an area element of .
Laplacian
The Laplacian of a scalar field is typically defined as the divergence of the gradient of as below:
Definition: Let be a scalar field. Then the Laplacian of is given by
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