91Ó°ÊÓ

Give precise mathematical definitions or descriptions of the concepts that follow. Then illustrate the definition with a graph or an algebraic example.

The integral of a function of two or three variables along a curve C

Make a chart of all the new notation, definitions, and theorems in this section, including what each new item means in terms you already understand.

Conservative Vector Fields: Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.

$F\left(x,y\right)=\frac{2x}{y}i+\frac{x^2}{y^2}j$.

Give two examples of quantities that may be computed by ${∫}_C \mathbf{F}â‹\dots d\mathbf{r}.$

Compute a general formula for $$dS$$ for any plane $$ax +by+cz = k$$ if $$c \neq 0$$.

For each integral in Exercises 5–8, give the vector field that is being integrated.

${∫}_C(x+y)\mathrm{d}x+xy\mathrm{d}y$

Calculus of vector-valued functions: Calculate each of the following.

$∫r\left(t\right)\mathrm{d}t,\mathrm{where}r\left(t\right)=e^t\mathbf{i}+t^3\mathbf{j}−4\mathbf{k}$

What are the outputs of a vector field in the Cartesian plane?

Suppose that S 1 is the upper half of the unit sphere, with outwards-pointing normal n1, and S 2 is a balloon-shaped surface whose boundary is the unit circle whose orientation leads to counterclockwise parametrization of the unit circle. If F(x, y ,z) is a smooth vector field defined on a region large enough to include both surfaces, what is the relationship between $∫{∫}_{{\mathrm{s}}_1}\mathrm{curl}\mathrm{F}.{\mathrm{n}}_1\mathrm{dS}\mathrm{and}∫{∫}_{{\mathrm{s}}_2}\mathrm{curl}\mathrm{F}.{\mathrm{n}}_2\mathrm{dS}?$

Compute a general formula for dS for any plane$ax+by+cz=kifc≠0.$