# Vector Integral Calculus

• ### Vector Integration

In the previous section, we discussed vector differentiation. We can extend our notion of integration of scalar functions to that of vector functions.

Let ${\vec F (x,y,z)}$ be a vector field defined over a region and let ${C}$ be a curve in this region. At each point ${\vec F}$ will have a value. We consider an arc element ${\delta s}$ and a unit tangent vector ${\hat T}$ at a point ${P}$ on the curve. If we fix 2 points ${A}$ and ${B}$ on the curve and allow ${P}$ to slide along the curve from ${A}$ to ${B}$, we get the path of integration.

${\int \limits_{C : A}^{B} \vec F \cdot \hat T ds}$

Above integral is known as the line integral.

Note that as ${\delta s \to 0}$, ${\hat T \delta s \to d \vec r}$. Hence, above integral can also be written as

${\int \limits_C \vec F \cdot d \vec r}$

• ### Conservative Field

If the definite integral ${\int \limits_{C : A}^{B} \vec F \cdot d \vec r}$ does not depend on the path ${C}$, the field ${\vec F}$ is known as conservative field. It can be shown that this is true, when there exists a scalar point function ${\phi}$, such that ${\vec F = \nabla \phi}$.

The closed path integral, where ${B}$ coincides with ${A}$, ${\oint \limits_C \vec F \cdot d \vec r}$, is ${0}$ in a conservative field.

• ### Gauss’ Divergence Theorem

The surface integral of a vector point function ${\vec F}$ over a surface ${S}$ is defined as the component of ${\vec F}$ normal to the surface ${S}$, taken over the entire surface. In other words,

${\int_S \vec F \cdot \hat n dS}$

is the surface integral. Since the elementary area ${dS}$ can be expressed as ${dx dy}$, the surface integral is actually a double integral.

Gauss’ Divergence Theorem states that the surface integral of the normal component of the function ${\vec F}$ taken over a closed surface ${S}$ is equal to the volume integral of ${\vec F}$ taken over the volume ${V}$ enclosed by the surface ${S}$. In other words,

${\iint \limits_S \vec F \cdot \hat n dS = \iiint \limits_V \nabla \cdot \vec F dV}$

• ### Stokes’ Theorem

The surface integral of curl of normal component of a vector point function ${\vec F}$ taken over an open surface ${S}$ bounded by a closed curve ${C}$ is equal to the line integral of the tangential component of ${\vec F}$ over ${C}$. Thus,

${\iint \limits_S (\nabla \times \vec F) \cdot \hat n dS = \oint \limits_C \vec F \cdot d \vec r}$

• ### Green’s Lemma

The Green’s lemma (or theorem) is a special case of Stokes’ theorem, when the surface is in ${XY}$ plane and thus ${\hat n = \hat k}$.