In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically I work out the Laplacian of the inverse of the magnitude of the separation vector. This is required in order to prove the Helmholtz Theorem

The Helmholtz theorem is vital to simplify the study of electric and magnetic fields; permitting the introduction of a scalar and magnetic potential. The proof of which is often overlooked, or in my opinion it's just shown to be true. The proof follows from the Dirac Delta Function (or distribution) which is how I do it. I prove all the necessary machinery required for the proof (which is quite involved in that regard).



Thanks to Andrew Weatherbee for pointing out a serious error in a previous version of this video.

1. Vector Components
2. Scalar Dot Product
3. Vector Cross Product (1/2)
4. Vector Cross Product (2/2)
5. Law of Cosines
6. Separation Vector
7. Nabla Operator (1/2)
8. The Gradient Grad
9. The Normal Vector
10. Why the Gradient is Perpendicular to Functions
11. Directional Derivative
12. The Nabla Operator (2/2)
13. The Divergence
14. The Curl of a Vector Field
15. Product Rules for Grad Div Curl
16. Vector Product Rule 2
17. Vector Product Rule 3
18. Vector Product Rule 4
19. Vector Product Rule 5
20. Vector Product Rule 6
21. Vector Quotient Rule 1
22. Vector Quotient Rule 2
23. Vector Quotient Rule 3
24. The Laplacian
25. Curl of the Gradient
26. Divergence of the Curl
27. Curl of the Curl
28. Fundamental Theorem of Calculus
29. Fundamental Theorem for Gradients
30. Green's Divergence Theorem
31. Stokes' Theorem
32. Integration by Parts Rule 1
33. Integration by Parts Example
34. Integration by Parts Rule 2
35. Integration by Parts Rule 3
36. Integration by Parts Rule 4
37. Spherical Polar Co-ordinates
38. Helmholtz Theorem (No Derivation)
39. Dirac Delta Function (1/2)
40. Dirac Delta Function (2/2)
41. Gradient of One Over the Separation Vector
42. Laplacian of One Over the Separation Vector
43. Helmholtz Theorem Proof (Part 1)
44. Helmholtz Theorem Proof (Part 2)
45. Derivation Biot and Savart Law

This video tutorial series covers a range of vector calculus topics such as grad, div, curl, the fundamental theorems, integration by parts, the Dirac Delta Function, the Helmholtz Theorem, spherical polar co-ordinates etc.

Thank you for watching and I hope that this matches your requirements. Please feel free to provide feedback via comments and share with your friends.

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