**Vector Calculus for Electromagnetism **is the backbone of any comprehensive treatment of Electromagnetism. This of course poses challenges to those seeking to understand the topic. None the less, while the vector calculus may seem complicated — it is quite straight forward in most cases. In fact, it’s probably fair to say that the mathematical tediousness heavily outweighs the actual conceptual difficulty. I have attempted to derive (in full) every expression, theorem or otherwise which you may require. I hope that this approach will greatly reduce your workload and simplify your studies!Return to Electromagnetism and Optics

Video # |
Video Tutorial Title |
Remarks |

1 | Vector Components | A quick revision of vector components in order to confirm the notation I’ll employ in future videos. |

2 | Scalar Dot Product | A quick revision discussing both its geometric meaning and its usefulness for calculating the flux of a field. |

3 a | Vector Cross Product 1/2 AxB |
Calculation of the vector cross product using matrices. |

3 b | Vector Cross Product 2/2 AxB |
The geometric interpretation of the vector cross product. |

4 | Law of Cosines | An extremely useful trick for analysing pairs of vectors. |

5 | The Separation Vector | This is the single most important vector in the study of electromagnetism.Every expression requires it as it is the vector difference between the source and detector co-ordinates. |

6 | The ‘Nabla’ Operator 1/2 ∇ | This operator is required in order to write Maxwell’s Equations in differential rather than integral form. |

7 | The Gradient ∇A |
The first of the three most useful operations for analysing vector fields.This operation is used to analyse rates of change. |

8 | The Normal Vector | A very fundamental quantity in analysing vector fields; the vector perpendicular to your field. |

9 | Why the Gradient is Perpendicular to Functions | This video discusses why the gradient of a vector field is ‘normal’ to the field. |

10 | The Directional Derivative | This analyses the rate of chance of your vector field in the direction of another specified vector. |

11 | The Nabla Operator 2/2 ∇ | Summary of the Nabla operator and its use for analysing vector fields (grad, div, curl). |

12 | The Divergence of a Vector Field ∇⋅A |
The second of the three most useful operations for analysing vector fields.This operation is used to analyse sources or sinks in the field. |

13 | The Curl of a Vector Field ∇xA |
The third of the three most useful operations for analysing vector fields.This operation is used to analyse rotations in the field. |

14 | Product Rules for Grad/Div/Curl | Fundamental vector calculus for the study of electromagnetism. The theory becomes extremely cumbersome without them. |

15 | Identities for Grad/Div/Curl | General identities for grad, div and curl. I need to make a PDF of this too |

16 | Product Rule 1 ∇(fA) |
Not uploaded |

17 | Product Rule 2 ∇⋅(fA) |
The Divergence of the product of a scalar and vector field. |

18 | Product Rule 3 ∇x(fA) |
The Curl of the product of a scalar and vector field. |

19 | Product Rule 4 ∇⋅(AxB) |
The Divergence of the vector field resulting from a cross product between two vector fields. |

20 | Product Rule 5 ∇x(AxB) |
The Curl of the vector field resulting from a cross product between two vector fields. |

21 | Product Rule 6 ∇(A⋅B) |
The Gradient of a scalar field resulting from a scalar product between two vector fields. |

22 | Quotient Rule 1 | |

23 | Quotient Rule 2 | |

24 | Quotient Rule 3 | |

25 | The Laplacian Operator | |

26 | Curl of the Gradient | |

27 | Divergence of the Curl | |

28 | Curl of the Curl | |

29 | Fundamental Theorem of Calculus | |

30 | Fundamental Theorem of Gradients | |

31 | Green’s Divergence Theorem | |

32 | Stokes’ Theorem | |

33 | Integration by Parts Rule 1 | A derivation of what most people refer to as integration by parts (line integrals) |

34 | Integration by Parts Example | |

35 | Integration by Parts Rule 2 | For volume and surface integrals |

36 | Integration by Parts Rule 3 | For surface and line integrals |

37 | Integration by Parts Rule 4 | For volume and surface integrals involving vector curls |

38 | Spherical Polar Co-ords | Converting rectangular to spherical co-ordinates |

39 | Helmholtz Theorem | No derivation; just a discussion of its importance |

40 | Dirac Delta Function 1/2 | Motivating the existence of the function/distribution |

41 | Dirac Delta Function 2/2 | Illustrating why its indispensable for physics |

42 | ∇(1/r) | Gradient of one over the Separation Vector |

43 | ∇^{2}(1/r) |
Laplacian of one over the Separation Vector |

44 | Helmholtz Theorem 1/2 | Derivation of the scalar potential |

45 | Helmholtz Theorem 2/2 | Derivation of the vector potential |

46 | Biot and Savart | Derivation of this fundamental magnetostatic equation |

47 | Laplacian in Spherical Co-ords | A very painful derivation of a most important and fundamental operator |

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