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}.

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