1. Tuesday Sept 21st: Vector Calculus Derivatives of a scalar field: gradient, directional derivative, Laplacian Derivatives of a vector field: divergence, curl

2. A homogeneous fluid on a rotating sphere Why we need it

3. A homogeneous fluid on a rotating sphere Why we need it vector divergence derivative cross product gradient unit vector Laplacian

4. Scalar fields no directionality, e.g. temperature, oxygen content Cartesian coordinates

5. e.g. surface pressure = P(longitude, latitude)

6. Differentiating a scalar: directional derivative, gradient gradient directional derivative

7. Pressure gradient Gradient is a VECTOR

8. Pressure gradient force

10. Examples

11. A vector differential operator “Del”, or “Nabla”,

12. The Laplacian Laplacian is a SCALAR

13. 2,3 dimensional PDEs Diffusion eq’n Wave eq’n

14. Vector fields e.g. velocity, acceleration, gradient

15. Differentiating a vector field examples

16. Divergence of a vector field Divergence is a SCALAR.

17. Curl Which way does the curl vector point? Example: river flow

18. Identities of vector calculus

20. Example: river flow Diffusion (friction) Concentration of velocity diffuses away

21. Example: river flow gravity

22. Example: river flow

23. curl

24. The Laplacian

25. Horizontal divergence

26. Modeling rain