6.1 Vector Fields

A vector field assigns a vector $F (x, y)$ to each point $(x, y)$ in a subset D of $ℝ^{2} or ℝ^{3} .$ $F (x, y, z)$ to each point $(x, y, z)$ in a subset D of $ℝ^{3} .$
Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.
We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.
A vector field $F$ is called conservative if there exists a scalar function $f$ such that $∇ f = F .$

6.2 Line Integrals

Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the x-axis, but the domain of integration in a line integral is a curve in a plane or in space.
If C is a curve, then the length of C is $\int_{C} d s .$
There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.
Scalar line integrals can be calculated using Equation 6.8; vector line integrals can be calculated using Equation 6.9.
Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.

The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.
The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.
Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.
Given vector field F, we can test whether F is conservative by using the cross-partial property. If F has the cross-partial property and the domain is simply connected, then F is conservative (and thus has a potential function). If F is conservative, we can find a potential function by using the Problem-Solving Strategy.
The circulation of a conservative vector field on a simply connected domain over a closed curve is zero.

Green’s theorem relates the integral over a connected region to an integral over the boundary of the region. Green’s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.
Green’s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is $F \cdot T .$ In the flux form, the integrand is $F \cdot N .$
Green’s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.
A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.

The divergence of a vector field is a scalar function. Divergence measures the “outflowing-ness” of a vector field. If v is the velocity field of a fluid, then the divergence of v at a point is the outflow of the fluid less the inflow at the point.
The curl of a vector field is a vector field. The curl of a vector field at point P measures the tendency of particles at P to rotate about the axis that points in the direction of the curl at P.
A vector field with a simply connected domain is conservative if and only if its curl is zero.

Surfaces can be parameterized, just as curves can be parameterized. In general, surfaces must be parameterized with two parameters.
Surfaces can sometimes be oriented, just as curves can be oriented. Some surfaces, such as a Möbius strip, cannot be oriented.
A surface integral is like a line integral in one higher dimension. The domain of integration of a surface integral is a surface in a plane or space, rather than a curve in a plane or space.
The integrand of a surface integral can be a scalar function or a vector field. To calculate a surface integral with an integrand that is a function, use Equation 6.19. To calculate a surface integral with an integrand that is a vector field, use Equation 6.20.
If S is a surface, then the area of S is $\int \int_{S} d S .$

Stokes’ theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes’ theorem is a higher dimensional version of Green’s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.
Stokes’ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.
Through Stokes’ theorem, line integrals can be evaluated using the simplest surface with boundary C.
Faraday’s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes’ theorem can be used to derive Faraday’s law.

The divergence theorem relates a surface integral across closed surface S to a triple integral over the solid enclosed by S. The divergence theorem is a higher dimensional version of the flux form of Green’s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus.
The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa.
The divergence theorem can be used to derive Gauss’ law, a fundamental law in electrostatics.