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

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

Scalar and vector fields

Let ${\displaystyle \mathbf {x} \,}$ 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).

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,

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

If ${\displaystyle {\boldsymbol {\nabla }}\cdot {\mathbf {u} }=0}$, 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 defined as

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

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

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

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

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