Vector analysis

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.

Classical Vector Analysis

Classical vector analysis largely is based developing the methods of calculus for $\mathbb{R}^3$ . 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 $f:U\to\mathbb{R}$ where $U$ is a subset of $\mathbb{R}^3$ . In other words, $f$ is a map that assigns a real-valued scalar to every point in $U$ . Generally, this scalar will represent some type of quantity such as potential in physics. Mainly, scalar fields are generalizations of functions with more variables: A function f(x) in $\mathbb{R}^2$ can be implicitly defined as a scalar field F(x, y) = x + f(x).

Vector fields

A vector field in the traditional sense is a map $\mathbf{F}:u\to\mathbb{R}^3$ where $U$ is a subset of $\mathbb{R}^3$ . That is, $F$ associates a vector in $\mathbb{R}^3$ to every point in $U$ . Typically, this construction is used to represent some kind of direction along with a quantity being associated with to a specific point in $\mathbb{R}^3$ . Vector fields are dimensionless relations respect to distance. This may be electric fields, magnetic fields, or even fields that model fluid flow ( $\mathbf{E}$ , $\mathbf{B}$ , $\mathbf{u}$ ). For a given vector field $\mathbf{F}$ , it has a flow of $\int_{C}\mathbf{F}\cdot\mathbf{t}\,ds$ and a flux $\Phi$ of $\iint_{E}\mathbf{F}\cdot\mathbf{n}\,dS$ .

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 measures how the vectors are "diverging out". If the divergence is positive, the vector field is considered a "source", and if it is negative, it is considered a "sink". If it is divergence-free, the vector field is incompressible. It is also the density of outward flux at a given point in $\mathbb{R}^3$ . Under this definition, divergence is often thought of as 'flux density' which is the motivation for the Divergence theorem.

Definition: Let $\mathbf{F}:U\to\mathbb{R}^3$ where $U$ is a subset of $\mathbb{R}^3$ . Then the divergence of $\mathbf{F}$ evaluated at the point $p\in U$ is given by $\text{div}(\mathbf{F})(p) = \nabla \cdot \mathbf{F} = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}\mathbf{F}\cdot\mathbf{n}\,dS,$ where $R$ is a region whose volume $V(R)$ shrinks to $0$ about the point $p$ , $\partial R$ is the boundary of $R$ , $\mathbf{n}$ is the outward unit normal of $\mathbf{F}$ relative to $R$ , and $dS$ is an area element of $R$ .

Curl

Curl of a vector field is the density of circulation at a given point in $\mathbb{R}^3$ . It can also be known as the "rotor" of the vector field, as it measures how the vectors are "spinning". Thus, one can think of curl intuitively as 'circulation density' which is motivation for Stokes' Theorem.

Definition: Let $\mathbf{F}:U\to\mathbb{R}^3$ where $U$ is a subset of $\mathbb{R}^3$ . Then the curl of $\mathbf{F}$ evaluated at the point $p\in U$ is given by $\text{curl}(\mathbf{F})(p)\cdot\mathbf{n} = \nabla \times \mathbf{F} \cdot\mathbf{n} = \lim_{A(R)\to 0} \frac{1}{A(R)}\int_{\partial R}\mathbf{F}\cdot\mathbf{t}\,ds,$ where $R$ is a region whose area $A(R)$ shrinks to $0$ about the point $p$ , $C$ is the boundary (closed loop) of $R$ , $\mathbf{n}$ is the outward unit normal of $\mathbf{F}$ relative to $R$ , $\mathbf{t}$ is the unit tangent of $\mathbf{F}$ relative to $R$ , and $ds$ is an arclength element of $C$ .

An alternative definition that one can give that does not rely upon the normal component of curl is as follows:

Definition: Let $\mathbf{F}:U\to\mathbb{R}^3$ where $U$ is a subset of $\mathbb{R}^3$ . Then the curl of $\mathbf{F}$ evaluated at the point $p\in U$ is given by $\text{curl}(\mathbf{F})(p) = \nabla \times \mathbf{F} = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}\mathbf{F}\times\mathbf{n}\,dS,$ where $R$ is a region whose volume $V(R)$ shrinks to $0$ about the point $p$ , $\partial R$ is the boundary of $R$ , $\mathbf{n}$ is the outward unit normal of $\mathbf{F}$ relative to $R$ , $\mathbf{t}$ is the unit tangent of $\mathbf{F}$ relative to $R$ , and $dS$ is an area element of $R$ .

Gradient

The gradient of a scalar field $f:\mathbb{R}^3\to\mathbb{R}$ is typically interpreted to be the operator that gives the vector whose direction is that in the direction of greatest increase of $f$ and magnitude being the rate at which $f$ is increasing. Gradient can be considered as the "derivative vector" of a given multivariable function. As such, we can define the gradient in a similar way to divergence and curls as below:

Definition: Let $f:U\to\mathbb{R}$ where $U$ is a subset of $\mathbb{R}^3$ . Then the gradient of $f$ evaluated at the point $p\in U$ is given by $\text{grad}(f)(p) = \nabla f = \lim_{V(R)\to 0} \frac{1}{V(R)}\iint_{\partial R}f\mathbf{n}\,dS,$ where $R$ is a region whose volume $V(R)$ shrinks to $0$ about the point $p$ , $\partial R$ is the boundary of $R$ , $\mathbf{n}$ is the outward unit normal relative to $R$ , and $dS$ is an area element of $R$ . The differential $df$ of a multivariable function is defined as $\nabla f \cdot d\mathbf{r}$ .

Laplacian

The Laplacian of a scalar field $f:\mathbb{R}^3\to\mathbb{R}$ is typically defined as the divergence of the gradient of $f$ as below:

Definition: Let $f:\mathbb{R}^3\to\mathbb{R}$ be a scalar field. Then the Laplacian of $f$ is given by $\text{div}(\text{grad}(f)) = \nabla \cdot \nabla f = f \nabla \cdot \nabla = \nabla^2f.$

The classical cases of Stokes' Theorem

In classical vector analysis, several integral theorems use Stoke's Theorem.

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