1.
Find the curl of \(\vec{F}=(x^2-z)\hat{i}+xe^z\hat{j}+xy\hat{k}\text{.}\) My preferred method uses determinants as opposed to memorizing the cumbersome formula. Citation 1
Answer.\(curl(\vec{F})=x(1-x^z)\hat{i}-(y+1)\hat{j}+e^z\hat{k}\)
Section 7.1 Curl of a Vector Field
Back when we first introduced Green's Theorem 4.1, we had a notion called circulation density.However, a little detail you might have missed is we also called this the "kth component of curl", without really expanding on what the full curl was. Now we see that. We are going to define the curl of a vector field and see what it means geometrically. We will also notice some nice properties that connected back to conservative vector fields.
Learning Objectives:
- Compute the curl of a vector field
- Interpret the curl geometrically
- Interpret the test for conservativeness of a field in terms of curl
TYPO: At 6:12 the kth component should be \(f_{yx}-f_{xy}\text{.}\)
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