Vector Calculus: Integrals, Coordinate Systems, and Theorems

17.2 - Vector Components

Vector Fields and Integrals

Vector calculus extends the concepts of integration and differentiation to vector fields, which are functions assigning a vector to every point in space. Understanding vector components is essential for evaluating integrals over curves and surfaces.

• Vector Field: A function F that assigns a vector to each point in space.

• Line Integral: The integral of a vector field along a curve $C$ is given by:

$\int_C \vec{F} \cdot d\vec{r}$

• Surface Integral: The integral of a vector field over a surface $S$:

$\iint_S \vec{F} \cdot d\vec{S}$

17.3 - Triplet Integrals

Triple Integrals and Volume

Triple integrals are used to compute volumes and other quantities in three-dimensional regions. The notation and limits depend on the coordinate system used.

• General Form:

$\iiint_E f(x, y, z) \, dV$

• Order of Integration: The order of $dx$, $dy$, $dz$ can vary depending on the region $E$.

• Applications: Calculating mass, charge, or other properties distributed in a volume.

17.4 - Coordinate Systems

Cylindrical and Spherical Coordinates

Changing coordinate systems can simplify the evaluation of integrals over symmetric regions.

• Cylindrical Coordinates: $(r, \theta, z)$

• Spherical Coordinates: $(\rho, \theta, \phi)$

System	Variables	Limits (Example)
Cylindrical	$r, \theta, z$	$0 \leq r \leq a$, $0 \leq \theta \leq 2\pi$, $0 \leq z \leq h$
Spherical	$\rho, \theta, \phi$	$0 \leq \rho \leq R$, $0 \leq \theta \leq 2\pi$, $0 \leq \phi \leq \pi$

17.5 - Line Integrals

Scalar and Vector Line Integrals

Line integrals generalize integration to functions along curves in space. They are used to compute work done by a force field, mass along a wire, and more.

• Scalar Line Integral:

$\int_C f(x, y, z) \, ds$

• Vector Line Integral:

$\int_C \vec{F} \cdot d\vec{r}$

• Applications: Work, circulation, and flux in physics and engineering.

17.6 - Surface Integrals

Integrals Over Surfaces

Surface integrals extend the concept of integration to two-dimensional surfaces in three-dimensional space. They are used to calculate flux through a surface.

• Surface Integral of a Scalar Function:

$\iint_S f(x, y, z) \, dS$

• Surface Integral of a Vector Field:

$\iint_S \vec{F} \cdot d\vec{S}$

17.7 - Divergence, Flux, and Theorems

Divergence, Flux, and Fundamental Theorems

Vector calculus includes several important theorems that relate integrals over regions, surfaces, and curves.

• Divergence: Measures the rate at which a vector field spreads out from a point.

$\nabla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$

• Flux: The total amount of a field passing through a surface.

$\text{Flux} = \iint_S \vec{F} \cdot d\vec{S}$

• Conservative Fields: A vector field is conservative if there exists a potential function $f$ such that $\vec{F} = \nabla f$.

• Gradient Theorem (Fundamental Theorem for Line Integrals):

$\int_C \nabla f \cdot d\vec{r} = f(B) - f(A)$

• Stokes' Theorem: Relates a surface integral of the curl of a vector field to a line integral around the boundary curve.

$\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \int_{\partial S} \vec{F} \cdot d\vec{r}$

• Divergence Theorem (Gauss' Theorem): Relates the flux of a vector field through a closed surface to the divergence over the volume inside.

$\iint_{\partial E} \vec{F} \cdot d\vec{S} = \iiint_E (\nabla \cdot \vec{F}) \, dV$

Coordinate Transformations

Changing Between Coordinate Systems

Transforming between rectangular, cylindrical, and spherical coordinates is essential for evaluating integrals over complex regions.

From	To	Transformation Equations
Rectangular	Cylindrical	$x = r \cos \theta$ $y = r \sin \theta$ $z = z$ $r = \sqrt{x^2 + y^2}$ $\theta = \tan^{-1}(y/x)$
Cylindrical	Rectangular	$x = r \cos \theta$ $y = r \sin \theta$ $z = z$
Rectangular	Spherical	$x = \rho \sin \phi \cos \theta$ $y = \rho \sin \phi \sin \theta$ $z = \rho \cos \phi$ $\rho = \sqrt{x^2 + y^2 + z^2}$ $\phi = \cos^{-1}(z/\rho)$
Spherical	Rectangular	$x = \rho \sin \phi \cos \theta$ $y = \rho \sin \phi \sin \theta$ $z = \rho \cos \phi$

Example: Volume of a Sphere Using Spherical Coordinates

• Set up the triple integral in spherical coordinates:

$\iiint_E dV = \int_0^{2\pi} \int_0^{\pi} \int_0^R \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$

• Evaluating gives the familiar formula for the volume of a sphere:

$V = \frac{4}{3} \pi R^3$

Additional info:

• These notes cover advanced topics in multivariable calculus, including vector calculus, coordinate transformations, and fundamental theorems. They are essential for students studying Calculus III or Vector Calculus.