Problem 1
In your own words, describe what an orientable surface is.
Problem 1
In the plane, flux is a measurement of how much of the vector field passes across a __________; in space, flux is ameasurement of how much of the vector field passes across a __________.
Problem 1
Give two quantities that can be represented by a vector field in the plane or in space.
Problem 2
How does the evaluation of a line integral given as \(\int_{c} f(s) d s\) differ from a line integral given as \(\oint_{C} f(s) d s ?\)
Problem 2
Let \(\vec{F}(x, y)\) be a vector field in the plane and let \(\vec{r}(t)\) be a two-dimensional vector-valued function. Why is \({ }^{\prime} \vec{F}(r(t))^{\prime \prime}\) an "abuse of notation"?
Problem 2
Give an example of a non-orientable surface.
Problem 2
In your own words, describe what it means for a vector field to have a negative divergence at a point.
Problem 2
When computing flux, what does it mean when the result is a negative number?
Problem 3
\(\mathrm{T} / \mathrm{F}:\) The orientation of a curve \(C\) matters when computing a line integral over a vector field.
Problem 3
What property of a vector field does Stokes' Theorem relate to circulation?
