### Key Concepts

### 6.1 Vector Fields

- A vector field assigns a vector $\text{F}(x,y)$ to each point $(x,y)$ in a subset
*D*of ${\mathbb{R}}^{2}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}{\mathbb{R}}^{3}.$ $\text{F}(x,y,z)$ to each point $(x,y,z)$ in a subset*D*of ${\mathbb{R}}^{3}.$ - 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 $\text{F}$ is called conservative if there exists a scalar function $f$ such that $\text{\u2207}f=\text{F}.$

### 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 ${\int}_{C}ds}.$ - 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.

### 6.3 Conservative Vector Fields

- 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.

### 6.4 Green’s Theorem

- 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 $\text{F}\xb7\text{T}.$ In the flux form, the integrand is $\text{F}\xb7\text{N}.$
- 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.

### 6.5 Divergence and Curl

- 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.

### 6.6 Surface Integrals

- 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 $\int {\displaystyle {\int}_{S}dS}}.$

### 6.7 Stokes’ Theorem

- 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.

### 6.8 The Divergence Theorem

- 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.