### Lecture Description

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.

### Course Index

- Vector Components
- Scalar Dot Product
- Vector Cross Product (1/2)
- Vector Cross Product (2/2)
- Law of Cosines
- Separation Vector
- Nabla Operator (1/2)
- The Gradient Grad
- The Normal Vector
- Why the Gradient is Perpendicular to Functions
- Directional Derivative
- The Nabla Operator (2/2)
- The Divergence
- The Curl of a Vector Field
- Product Rules for Grad Div Curl
- Vector Product Rule 2
- Vector Product Rule 3
- Vector Product Rule 4
- Vector Product Rule 5
- Vector Product Rule 6
- Vector Quotient Rule 1
- Vector Quotient Rule 2
- Vector Quotient Rule 3
- The Laplacian
- Curl of the Gradient
- Divergence of the Curl
- Curl of the Curl
- Fundamental Theorem of Calculus
- Fundamental Theorem for Gradients
- Green's Divergence Theorem
- Stokes' Theorem
- Integration by Parts Rule 1
- Integration by Parts Example
- Integration by Parts Rule 2
- Integration by Parts Rule 3
- Integration by Parts Rule 4
- Spherical Polar Co-ordinates
- Helmholtz Theorem (No Derivation)
- Dirac Delta Function (1/2)
- Dirac Delta Function (2/2)
- Gradient of One Over the Separation Vector
- Laplacian of One Over the Separation Vector
- Helmholtz Theorem Proof (Part 1)
- Helmholtz Theorem Proof (Part 2)
- Derivation Biot and Savart Law

### Course Description

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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