# Vector calculus

Material from Vectors was moved[1] here.

Here we extend the concept of vector to that of the vector field. A familiar example of a vector field is wind velocity: It has direction and magnitude, which makes it a vector. But it also depends on position (and ultimately on time). Wind velocity is a function of (x,y,z) at any given time, equivalently we can say that wind velocity is a time-dependent field: ${\displaystyle {\vec {V}}_{wind}={\vec {V}}({\vec {r}},t)}$.

### Derivative of a vector valued function

Let ${\displaystyle \mathbf {a} (x)\,}$ be a vector function that can be represented as

${\displaystyle \mathbf {a} (x)=a_{1}(x)\mathbf {e} _{1}+a_{2}(x)\mathbf {e} _{2}+a_{3}(x)\mathbf {e} _{3}\,}$

where ${\displaystyle x\,}$ is a scalar.

Then the derivative of ${\displaystyle \mathbf {a} (x)\,}$ with respect to ${\displaystyle x\,}$ is

${\displaystyle {\cfrac {d\mathbf {a} (x)}{dx}}=\lim _{\Delta x\rightarrow 0}{\cfrac {\mathbf {a} (x+\Delta x)-\mathbf {a} (x)}{\Delta x}}={\cfrac {da_{1}(x)}{dx}}\mathbf {e} _{1}+{\cfrac {da_{2}(x)}{dx}}\mathbf {e} _{2}+{\cfrac {da_{3}(x)}{dx}}\mathbf {e} _{3}~.}$

Note: In the above equation, the unit vectors ${\displaystyle \mathbf {e} _{i}}$ (i=1,2,3) are assumed constant.
If ${\displaystyle \mathbf {a} (x)\,}$ and ${\displaystyle \mathbf {b} (x)\,}$ are two vector functions, then from the chain rule we get

{\displaystyle {\begin{aligned}{\cfrac {d({\mathbf {a} }\cdot {\mathbf {b} })}{x}}&={\mathbf {a} }\cdot {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\cdot {\mathbf {b} }\\{\cfrac {d({\mathbf {a} }\times {\mathbf {b} })}{dx}}&={\mathbf {a} }\times {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\times {\mathbf {b} }\\{\cfrac {d[{\mathbf {a} }\cdot {({\mathbf {a} }\times {\mathbf {b} })}]}{dt}}&={\cfrac {d\mathbf {a} }{dt}}\cdot {({\mathbf {b} }\times {\mathbf {c} })}+{\mathbf {a} }\cdot {\left({\cfrac {d\mathbf {b} }{dt}}\times {\mathbf {c} }\right)}+{\mathbf {a} }\cdot {\left({\mathbf {b} }\times {\cfrac {d\mathbf {c} }{dt}}\right)}\end{aligned}}}

### Scalar and vector fields

Let ${\displaystyle \mathbf {x} \,}$ be the position vector of any point in space. Suppose that there is a scalar function (${\displaystyle g\,}$) that assigns a value to each point in space. Then

${\displaystyle g=g(\mathbf {x} )\,}$

represents a scalar field. An example of a scalar field is the temperature. See Figure4(a).

Figure 4: Scalar and vector fields.

If there is a vector function (${\displaystyle \mathbf {a} \,}$) that assigns a vector to each point in space, then

${\displaystyle \mathbf {a} =\mathbf {a} (\mathbf {x} )\,}$

represents a vector field. An example is the displacement field. See Figure 4(b).

### Gradient of a scalar field

Let ${\displaystyle \varphi (\mathbf {x} )\,}$ be a scalar function. Assume that the partial derivatives of the function are continuous in some region of space. If the point ${\displaystyle \mathbf {x} \,}$ has coordinates (${\displaystyle x_{1},x_{2},x_{3}\,}$) with respect to the basis (${\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\,}$), the gradient of ${\displaystyle \varphi \,}$ is defined as

${\displaystyle {\boldsymbol {\nabla }}{\varphi }={\frac {\partial \varphi }{\partial x_{1}}}~\mathbf {e} _{1}+{\frac {\partial \varphi }{\partial x_{2}}}~\mathbf {e} _{2}+{\frac {\partial \varphi }{\partial x_{3}}}~\mathbf {e} _{3}~.}$

In index notation,

${\displaystyle {\boldsymbol {\nabla }}{\varphi }\equiv \varphi _{,i}~\mathbf {e} _{i}~.}$

The gradient is obviously a vector and has a direction. We can think of the gradient at a point being the vector perpendicular to the level contour at that point.

It is often useful to think of the symbol ${\displaystyle {\boldsymbol {\nabla }}{}}$ as an operator of the form

${\displaystyle {\boldsymbol {\nabla }}{}={\frac {\partial }{\partial x_{1}}}~\mathbf {e} _{1}+{\frac {\partial }{\partial x_{2}}}~\mathbf {e} _{2}+{\frac {\partial }{\partial x_{3}}}~\mathbf {e} _{3}~.}$

### Divergence of a vector field

If we form a scalar product of a vector field ${\displaystyle \mathbf {u} (\mathbf {x} )\,}$ with the ${\displaystyle {\boldsymbol {\nabla }}{}}$ operator, we get a scalar quantity called the divergence of the vector field. Thus,

${\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {u} ={\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{3}}{\partial x_{3}}}~.}$

In index notation,

${\displaystyle {\boldsymbol {\nabla }}\mathbf {u} \equiv u_{i,i}~.}$

If ${\displaystyle {\boldsymbol {\nabla }}\mathbf {u} =0}$, then ${\displaystyle \mathbf {u} \,}$ is called a divergence-free field.

The physical significance of the divergence of a vector field is the rate at which some density exits a given region of space. In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.

### Curl of a vector field

The curl of a vector field ${\displaystyle \mathbf {u} (\mathbf {x} )\,}$ is a vector whose expression can be obtained with

${\displaystyle {\boldsymbol {\nabla }}\times {\mathbf {u} }={\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\{\frac {\partial }{\partial x_{1}}}&{\frac {\partial }{\partial x_{2}}}&{\frac {\partial }{\partial x_{3}}}\\u_{1}&u_{2}&u_{3}\\\end{vmatrix}}}$

The physical significance of the curl of a vector field is the amount of rotation or angular momentum of the contents of a region of space.

### Laplacian of a scalar or vector field

The Laplacian of a scalar field ${\displaystyle \varphi (\mathbf {x} )\,}$ is a scalar defined as

${\displaystyle \nabla ^{2}{\varphi }:={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}{\varphi })={\frac {\partial ^{2}\varphi }{\partial x_{1}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{3}}}~.}$

The Laplacian of a vector field ${\displaystyle \mathbf {u} (\mathbf {x} )\,}$ is a vector defined as

${\displaystyle \nabla ^{2}{\mathbf {u} }:=(\nabla ^{2}{u_{1}})\mathbf {e} _{1}+(\nabla ^{2}{u_{2}})\mathbf {e} _{2}+(\nabla ^{2}{u_{3}})\mathbf {e} _{3}~.}$

### Identities in vector calculus

Some frequently used identities from vector calculus are listed below.

1. ${\displaystyle {\boldsymbol {\nabla }}(\mathbf {a} +\mathbf {b} )={\boldsymbol {\nabla }}\cdot {\mathbf {a} }+{\boldsymbol {\nabla }}\cdot {\mathbf {b} }}$
2. ${\displaystyle {\boldsymbol {\nabla }}\times {(\mathbf {a} +\mathbf {b} )}={\boldsymbol {\nabla }}\times {\mathbf {a} }+{\boldsymbol {\nabla }}\times {\mathbf {b} }}$
3. ${\displaystyle {\boldsymbol {\nabla }}(\varphi \mathbf {a} )=\cdot {({\boldsymbol {\nabla }}{\varphi })}{\mathbf {a} }+\varphi ({\boldsymbol {\nabla }}\cdot {\mathbf {a} })}$
4. ${\displaystyle {\boldsymbol {\nabla }}\times {(\varphi \mathbf {a} )}={({\boldsymbol {\nabla }}{\varphi })}\times {\mathbf {a} }+\varphi ({\boldsymbol {\nabla }}\times {\mathbf {a} })}$
5. ${\displaystyle {\boldsymbol {\nabla }}({\mathbf {a} }\times {\mathbf {b} })={\mathbf {b} }\cdot {({\boldsymbol {\nabla }}\times {\mathbf {a} })}-{\mathbf {a} }\cdot {({\boldsymbol {\nabla }}\times {\mathbf {b} })}}$

## Fundamental theorems of vector calculus

One version of the fundamental theorem of one-dimensional calculus is

${\displaystyle \int _{a}^{b}f\,'(x)dx=f(b)-f(a)}$

This is a theorem about a function, ${\displaystyle f(x)}$, its first derivative, and a line segment. Two notations used to denote this line segment are [a,b] and the inequality, a<x<b. In the field of topology, ${\displaystyle \partial }$ denotes boundary. If we let the symbol ${\displaystyle {\mathcal {L}}}$ denote the infinite number of points in the line segment [a,b], then the symbol ${\displaystyle \partial {\mathcal {L}}}$ denotes the two endpoints (at x = a and x = b ) of the line segment ${\displaystyle {\mathcal {L}}}$. These endpoints form the boundary of ${\displaystyle {\mathcal {L}}}$.

The the gradient theorem is a direct generalization of the fundamental theorem of calculus:

${\displaystyle \int _{\ell [{\vec {p}}\to {\vec {q}}]\subset \mathbb {R} ^{n}}{\vec {\nabla }}f\cdot d{\vec {\ell }}=f\left({\vec {q}}\right)-f\left({\vec {p}}\right)}$

The subscript, ${\displaystyle \ell [{\vec {p}}\to {\vec {q}}]\subset \mathbb {R} ^{n}}$ informs this is an integral over the over a one-dimensinal curve (or 'path') line integral ${\displaystyle \ell }$ from point ${\displaystyle {\vec {r}}={\vec {p}}}$ to point ${\displaystyle {\vec {r}}={\vec {q}}}$. The function, ${\displaystyle f=f({\vec {r}})}$ is any scalar field that is differentiable. The expression ${\displaystyle \subset \mathbb {R} ^{n}}$ informs us that ${\displaystyle {\vec {r}}}$ can be a member of an n-dimensional space. (In other words the theorem is easily generalized to more than three dimensions.) A consequence of this theorem is that ${\displaystyle \int {\vec {\nabla }}f\cdot d{\vec {\ell }}=0}$ for any "closed curve" The figure shows the closed curve A, as well as the "open curve", B. Two endpoints form the "boundary" of curve B.

### Stokes' theorem

Stokes' theorem states:

${\displaystyle \int _{\Sigma \subset \mathbb {R} ^{3}}{\vec {\nabla }}\times {\vec {F}}\cdot {\vec {dA}}=\oint _{\partial \Sigma }{\vec {F}}\cdot d{\hat {\ell }}}$

The integral subscript, ${\displaystyle \Sigma \subset \mathbb {R} ^{3}}$ informs us that this theorem is valid only in a three-dimensional vector space. The integral is over a two-dimensional surface,Σ ,with ${\displaystyle {\vec {dA}}={\hat {n}}dA}$, where ${\displaystyle {\hat {n}}}$ is normal to the surface. The integral over the surface, Σ, is nonzero only if its boundary, ∂Σ, exists. Surfaces with such boundaries are called open surfaces, and the boundary, ∂Σ, is a curve in 3-space that goes along the "edge" of the surface. This curve is integrated in the direction of positive orientation, meaning that ${\displaystyle {\vec {d\ell }}}$ and the surface normal follow ${\displaystyle {\hat {n}}dA}$ follow the right-hand rule.

Footnote: According to Wikipedia[2], this form of the theorem was first discovered by Lord Kelvin, who communicated it to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name.

### Divergence theorem

closed  | open: Ω & ∂Ω

The divergence theorem states:

${\displaystyle \int _{\Omega }{\vec {\nabla }}\cdot {\vec {F}}\;dV=\oint _{\partial \Omega \subset \mathbb {R} ^{3}}{\vec {F}}\cdot d{\vec {A}}}$.

The integral subscript, ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$, informs us that this theorem is valid in an (arbitrary) n-dimensional vector space. The n-dimensional volume is Ω, and ∂Ω is its boundary. If n =3 dimensions, ∂Ω is a surface. Since this surface encloses a volume, it has no boundary of its own, and is therefore called a closed surface. The figure shows six surfaces. The three on the left have no boundary and are therefore closed; the ones to the right have a boundary (shown in red) and are therefore open. Note that the closed surfaces to the left are themselves boundaries volumes which are defined as what is "inside" the surface.

Footnote: In index notation, the gradient theorem can be written as ${\displaystyle \int _{\Omega }u_{i,i}~dV=\int _{\partial \Omega }n_{i}u_{i}~dA}$