Vector Integration And Integral Theorems

1. Core Concept Overview & Intuition

Vector integration extends the ordinary definite integral from scalar functions of one variable to vector fields evaluated along curves, over surfaces, and throughout solid regions. The central object is a vector field $\vec{F}(x,y,z)=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$, which assigns a vector to every point of a region in space. Integration of such a field requires a path, a surface, or a volume over which the field is sampled, and the result depends on the geometric nature of that domain.

Three families of integrals arise. A line integral accumulates the tangential component of $\vec{F}$ along a curve $C$; when $\vec{F}$ represents a force, this integral is the work done by the force on a particle traversing $C$. A surface integral accumulates either the normal component of $\vec{F}$ across a surface $S$ (a flux) or the value of a scalar field over $S$ (an area-type integral). A volume integral accumulates a field throughout a closed region $V$.

The integral theorems convert one type of integral into another. The divergence theorem converts a flux integral over a closed surface into a volume integral of the divergence. Green's theorem converts a line integral around a plane curve into a double integral over the enclosed plane region. Stokes' theorem converts a line integral around a closed space curve into a surface integral of the curl over any surface spanning that curve. These conversions are the principal computational strategy of the topic: a difficult integral is replaced by an easier one of a different type.

A second organizing idea is path independence. When the field is the gradient of a scalar potential, $\vec{F}=\nabla\phi$, the line integral depends only on the endpoints and not on the route, and the integral around any closed curve vanishes. This connects the integral calculus of this topic to the differential operators $\nabla$, $\nabla\cdot$, and $\nabla\times$ studied earlier.

2. Mathematical Formalization & Core Definitions

2.1 Parametrized Curves and the Line Integral

> Definition (Smooth curve). Let $\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$ be the position vector of the point $(x,y,z)$, defining a curve $C$ joining points $P_1$ and $P_2$ where $t=t_1$ and $t=t_2$ respectively. The curve $C$ is composed of a finite number of curves for each of which $\vec{r}(t)$ has a continuous derivative.

> Definition (Line integral of a vector field). Let $\vec{F}(x,y,z)=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$ be a vector function of position defined and continuous along $C$. The integral of the tangential component of $\vec{F}$ along $C$ from $P_1$ to $P_2$ is

$$
\int_{P_1}^{P_2}\vec{F}\cdot d\vec{r}=\int_C\vec{F}\cdot d\vec{r}.
$$

> If $\vec{F}$ is a force on a particle moving along $C$, this line integral represents the work done by the force. If $C$ is a simple closed curve, the integral around $C$ is often denoted $\oint_C\vec{F}\cdot d\vec{r}$. In general, any integral evaluated along a curve is called a line integral.

In coordinates, with $d\vec{r}=dx\,\hat{i}+dy\,\hat{j}+dz\,\hat{k}$ and $\vec{F}=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$, the line integral takes the scalar form

$$
\int_C\vec{F}\cdot d\vec{r}=\int_C\left(F_1\,dx+F_2\,dy+F_3\,dz\right).
$$

For a plane field $\vec{F}=M\hat{i}+N\hat{j}$ this reduces to $\int_C(M\,dx+N\,dy)$.

2.2 Evaluation Technique for Line Integrals

The standard procedure is to substitute the parametrization of $C$ into the integrand and integrate in the single parameter $t$. When $C$ is a polygonal path, the integral is split additively over the constituent segments, and on each segment the constant coordinate is used to eliminate one differential. For a segment on which $y$ is constant, $dy=0$; on a segment on which $x$ is constant, $dx=0$. Reversing the direction of traversal reverses the sign of the integral.

2.3 Surface Integrals

> Definition (Vector surface element). For a plane region in the $xy$-plane, the vector element of area is $d\vec{S}=dx\,dy\,\hat{n}$, where $\hat{n}$ is the unit vector normal to the plane made by the $\hat{i}$ and $\hat{j}$ directions. For a general surface $S$ in space, $d\vec{S}=ds\,\hat{n}$, where $ds$ is an element of area of $S$ and $\hat{n}$ is the unit normal to $S$.

> Definition (Surface integrals). For $\vec{F}$ a vector field and $\phi$ a scalar field, the integrals

$$
\iint_S\vec{F}\cdot d\vec{S}=\iint_S\vec{F}\cdot\hat{n}\,ds,\qquad \iint_S\phi\,dS=\iint_S\phi\,ds,\qquad \iint_S\phi\,\hat{n}\,ds,\qquad \iint_S\vec{F}\times d\vec{S}
$$

> are examples of surface integrals.

To evaluate surface integrals it is convenient to express them as double integrals taken over the projected area of $S$ on one of the coordinate planes. For a smooth surface given by $z=f(x,y)$ defined over a region $R_{xy}$ in the $xy$-plane, one writes $F(x,y,z)=f(x,y)-z=0$ and regards $S$ as the level surface $F(x,y,z)=0$. The unit normal is $\hat{n}=\nabla F/\lVert\nabla F\rVert$, and the surface element satisfies

$$
dS=\frac{dA}{\lVert\nabla F\rVert\cdot\lvert\hat{n}\cdot\hat{k}\rvert},
$$

so that the area of the surface is

$$
S=\iint_{R_{xy}}dA=\iint_{R_{xy}}\sqrt{f_x^2+f_y^2+1}\;dx\,dy.
$$

Analogous formulations hold for a smooth surface $x=f(y,z)$ over a region $R_{yz}$ and for $y=f(x,z)$ over a region $R_{xz}$.

2.4 Volume Integrals

> Definition (Volume integral). If $V$ is a closed region bounded by a surface $S$ and $\vec{F}$ is a vector field at each point of $V$ and on its boundary surface $S$, then

$$
\int_V\vec{F}\,dV=\iiint_V\vec{F}\,dV
$$

> is the volume integral of $\vec{F}$ throughout the region. Triple integral notation is standard for volume integrals.

In Cartesian coordinates $dV=dx\,dy\,dz$, and the vector integral is evaluated component by component.