Calculus Volume 3 · free preview
Divergence and Curl
Divergence measures a field's local outflow per volume, curl its local rotation, the two derivatives the vector field's basic differential operators with the theorems of the coming sections as their integral identities.
1. Divergence
Divergence
2. Curl
Curl
3. The vector identities
The vector identities
4. The Helmholtz decomposition
The Helmholtz decomposition
5. Divergence definition
The divergence of F = (P, Q, R) equals which expression?
6. Curl definition
The curl of F = (P, Q, R) has which z component?
7. Divergence of curl
The divergence of the curl of any smooth field equals what?
8. Curl of gradient
The curl of the gradient of a scalar function equals what?
9. Irrotational meaning
A field with zero curl everywhere is called irrotational. What does this imply for its flow?
10. Divergence radial
The divergence of (x, y, z) is 3. As a number, what is it? Answer with a number.
11. Curl rotation z
The curl of (-y, x, 0) is (0, 0, 2). As a number, what is the z component? Answer with a number.
12. Incompressible meaning
A zero divergence field is called incompressible, conserving which quantity? Answer with one word.
13. Nabla symbol
The operator symbol for divergence and curl is the inverted delta, nabla. The curl is nabla cross F. As an operation symbol between nabla and F, what is the divergence? Answer with the single word.
14. Laplacian name
The divergence of the gradient is the Laplacian. As a surname, whose? Answer in lowercase.
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