Vectors/Vector Calculus

Vector Calculus

So far we have dealt with constant vectors. It also helps if the vectors are allowed to vary in space. Then we can define derivatives and integrals and deal with vector fields. Some basic ideas of vector calculus are discussed below.

Derivative of a vector valued function

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

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

where $x\,$ is a scalar.

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

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

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

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

Scalar and vector fields

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

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

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

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

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

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

Gradient of a scalar field

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

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

In index notation,

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

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 ${\boldsymbol {\nabla }}{}$ as an operator of the form

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

Divergence of a vector field

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

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

In index notation,

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

If ${\boldsymbol {\nabla }}\cdot {\mathbf {u} }=0$, then $\mathbf {u} \,$ 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 $\mathbf {u} (\mathbf {x} )\,$ is a vector defined as

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

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 $\varphi (\mathbf {x} )\,$ is a scalar defined as

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

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

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

Identities in vector calculus

Some frequently used identities from vector calculus are listed below.

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

Green-Gauss Divergence Theorem

Let $\mathbf {u} (\mathbf {x} )\,$ be a continuous and differentiable vector field on a body $\Omega \,$ with boundary $\Gamma \,$. The divergence theorem states that

${\int _{\Omega }{\boldsymbol {\nabla }}\cdot {\mathbf {u} }~dV=\int _{\Gamma }{\mathbf {n} }\cdot {\mathbf {u} }~dA}$

where $\mathbf {n} \,$ is the outward unit normal to the surface (see Figure 5).

In index notation,

$\int _{\Omega }u_{i,i}~dV=\int _{\Gamma }n_{i}u_{i}~dA$

For more details on the topics of this chapter, see Vector calculus in the wikibook on Calculus.