# Vector Calculus for Electromagnetism

## Video Lectures

Displaying all 45 video lectures.

Lecture 1Play Video |
Vector ComponentsIn 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 discuss the components which we use as a basis in the rectangular coordinate system. |

Lecture 2Play Video |
Scalar Dot ProductIn 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 discuss the dot or scalar product which is an essential quantity for simplifying mathematical expressions. |

Lecture 3Play Video |
Vector Cross Product (1/2)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 discuss the vector produce which is an essential quantity for simplifying complicated mathematical expressions. Furthermore, it's physical interpretation aides in the analysis of the physics! |

Lecture 4Play Video |
Vector Cross Product (2/2)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 discuss the vector produce which is an essential quantity for simplifying complicated mathematical expressions. Furthermore, it's physical interpretation aides in the analysis of the physics! |

Lecture 5Play Video |
Law of CosinesIn 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 derive the Law of Cosines. This is a very useful quantity when analysing vector function using two separate vectors. In the study of electrodynamics, many vector functions involve two separate vectors (usually the separation vector) and thus this expression greatly simplifies the analysis! |

Lecture 6Play Video |
Separation VectorIn 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 discuss the separation vector. This vector is the vector difference between the a vector to the sources of a field and the vector to our detector. It is used in every expression in electrodynamics. |

Lecture 7Play Video |
Nabla Operator (1/2)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 introduce arguably the most important operator in vector calculus. |

Lecture 8Play Video |
The Gradient GradIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in introduce the first of the uses of the Nable operator. The gradient is fundamental to vector calculus and any studies of electromagnetism. |

Lecture 9Play Video |
The Normal VectorIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss how we calculate the vector which is perpendicular to or 'normal' to a function. This is important for Lagrange Multipliers and our studies of electromagnetism. |

Lecture 10Play Video |
Why the Gradient is Perpendicular to FunctionsIn 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 build upon video number 8 where I discussed the normal vector. In fact we see that the gradient calculates the normal vector for us. |

Lecture 11Play Video |
Directional DerivativeIn 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 discuss how we calculate the vector which calculates the rate of change of a function in a specific direction. |

Lecture 12Play Video |
The Nabla Operator (2/2)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 in this video I summarise our knowledge of the Nabla operator and prepare for its use in our studies of electromagnetism. |

Lecture 13Play Video |
The DivergenceIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss the divergence of a vector field. This mathematical construct is used both because it has a very interesting physical interpretation and that it simplifies complicated mathematical expressions. |

Lecture 14Play Video |
The Curl of a Vector FieldIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss the curl of a vector field both because it has a very important physical interpretation that it greatly simplifies complicated mathematical expressions. |

Lecture 15Play Video |
Product Rules for Grad Div CurlIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss the vector produce rules for the gradient, divergence and curl. These allow us to write Maxwell's Equations in a simpler fashion than would other wise be possible. |

Lecture 16Play Video |
Vector Product Rule 2In 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 discuss the divergence of a function which is the product of a scalar and a vector. |

Lecture 17Play Video |
Vector Product Rule 3In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the curl of a function which is a product of both a vector and a scalar. |

Lecture 18Play Video |
Vector Product Rule 4In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the divergence of a cross product. |

Lecture 19Play Video |
Vector Product Rule 5In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the curl of a cross product. |

Lecture 20Play Video |
Vector Product Rule 6In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the gradient of a scalar dot product. |

Lecture 21Play Video |
Vector Quotient Rule 1In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the gradient of a scalar quotient. |

Lecture 22Play Video |
Vector Quotient Rule 2In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the divergence of a function which is a quotient of a vector and a scalar. |

Lecture 23Play Video |
Vector Quotient Rule 3In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the curl of a function which is the quotient of a vector and a scalar. |

Lecture 24Play Video |
The LaplacianIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I introduce yet another important operator for our studies of electromagnetism. |

Lecture 25Play Video |
Curl of the GradientIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the curl of the grad of a vector field. |

Lecture 26Play Video |
Divergence of the CurlIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when we take the divergence of the curl of a vector field. |

Lecture 27Play Video |
Curl of the CurlIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss what happens when you take the double curl of a vector function. |

Lecture 28Play Video |
Fundamental Theorem of CalculusIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. |

Lecture 29Play Video |
Fundamental Theorem for GradientsIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I extend the Fundamental Theorem of Calculus to three dimensions. |

Lecture 30Play Video |
Green's Divergence TheoremIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss the fundamental theorem for divergences. |

Lecture 31Play Video |
Stokes' TheoremIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss the fundamental theorem for curls. |

Lecture 32Play Video |
Integration by Parts Rule 1In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss an extremely powerful technique in mathematics known as integration by parts. We shall see in the future that it can be done in many may ways on very many expressions! |

Lecture 33Play Video |
Integration by Parts ExampleIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss an extremely powerful technique in mathematics known as integration by parts. We shall see in the future that it can be done in many may ways on very many expressions! |

Lecture 34Play Video |
Integration by Parts Rule 2In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss an extremely powerful technique in mathematics known as integration by parts. We shall see in the future that it can be done in many may ways on very many expressions! |

Lecture 35Play Video |
Integration by Parts Rule 3In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss an extremely powerful technique in mathematics known as integration by parts. We shall see in the future that it can be done in many may ways on very many expressions! |

Lecture 36Play Video |
Integration by Parts Rule 4In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss an extremely powerful technique in mathematics known as integration by parts. We shall see in the future that it can be done in many may ways on very many expressions! |

Lecture 37Play Video |
Spherical Polar Co-ordinatesIn this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level. Specifically in this video I discuss transforming from rectangular to spherical polar co-ordinates. This greatly simplifies calculations when there exists spherical symmetry. |

Lecture 38Play Video |
Helmholtz Theorem (No Derivation)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 in this video I discuss but I don't derive the Helmholtz Theorem. This theorem allows the use of scalar and vector potentials which are the backbone of the study of electromagnetism. See later videos in this section for its derivation (numbers 44 and 45). |

Lecture 39Play Video |
Dirac Delta Function (1/2)In this video I continue my videos on Vector Calculus For Electromagnetism. Specifically I discuss the Dirac Delta Function (part 2 is the next video). This is vital for Electromagnetism as it permits the use of the Helmholtz Theorem and therefore the electric and magnetic potentials (vector and scalar). |

Lecture 40Play Video |
Dirac Delta Function (2/2)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 discuss the Dirac Delta Function (part 2 is the next video). This is vital for Electromagnetism as it permits the use of the Helmholtz Theorem and therefore the electric and magnetic potentials (vector and scalar). |

Lecture 41Play Video |
Gradient of One Over the Separation VectorIn 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 gradient 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). |

Lecture 42Play Video |
Laplacian of One Over the Separation VectorIn 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. |

Lecture 43Play Video |
Helmholtz Theorem Proof (Part 1)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 derive the scalar potential for 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). |

Lecture 44Play Video |
Helmholtz Theorem Proof (Part 2)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 derive the vector potential for 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). |

Lecture 45Play Video |
Derivation Biot and Savart LawIn 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 derive the Biot and Savart Law for magnetism. In many textbooks it's noted as a law derived from experiment. However, it's a consequence of Maxwell's equations (as is everything in this regard too!). It requires some tricky vector calculus and that's why it's in my vector calculus tutorials and not my magnetostatics section. |