Vector Calculus for Electromagnetism

Video Lectures

Displaying all 45 video lectures.
Lecture 1
Vector Components
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Vector Components
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 components which we use as a basis in the rectangular coordinate system.
Lecture 2
Scalar Dot Product
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Scalar Dot Product
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 dot or scalar product which is an essential quantity for simplifying mathematical expressions.
Lecture 3
Vector Cross Product (1/2)
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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 4
Vector Cross Product (2/2)
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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 5
Law of Cosines
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Law of Cosines
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 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 6
Separation Vector
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Separation Vector
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 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 7
Nabla Operator (1/2)
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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 8
The Gradient Grad
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The Gradient Grad
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 introduce the first of the uses of the Nable operator. The gradient is fundamental to vector calculus and any studies of electromagnetism.
Lecture 9
The Normal Vector
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The Normal Vector
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 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 10
Why the Gradient is Perpendicular to Functions
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Why the Gradient is Perpendicular to Functions
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 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 11
Directional Derivative
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Directional Derivative
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 how we calculate the vector which calculates the rate of change of a function in a specific direction.
Lecture 12
The Nabla Operator (2/2)
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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 13
The Divergence
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The Divergence
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 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 14
The Curl of a Vector Field
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The Curl of a Vector Field
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 the curl of a vector field both because it has a very important physical interpretation that it greatly simplifies complicated mathematical expressions.
Lecture 15
Product Rules for Grad Div Curl
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Product Rules for Grad Div Curl
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 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 16
Vector Product Rule 2
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Vector Product Rule 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 divergence of a function which is the product of a scalar and a vector.
Lecture 17
Vector Product Rule 3
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Vector Product Rule 3
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 what happens when we take the curl of a function which is a product of both a vector and a scalar.
Lecture 18
Vector Product Rule 4
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Vector Product Rule 4
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 what happens when we take the divergence of a cross product.
Lecture 19
Vector Product Rule 5
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Vector Product Rule 5
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 what happens when we take the curl of a cross product.
Lecture 20
Vector Product Rule 6
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Vector Product Rule 6
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 what happens when we take the gradient of a scalar dot product.
Lecture 21
Vector Quotient Rule 1
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Vector Quotient Rule 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 in this video I discuss what happens when we take the gradient of a scalar quotient.
Lecture 22
Vector Quotient Rule 2
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Vector Quotient Rule 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 discuss what happens when we take the divergence of a function which is a quotient of a vector and a scalar.
Lecture 23
Vector Quotient Rule 3
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Vector Quotient Rule 3
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 what happens when we take the curl of a function which is the quotient of a vector and a scalar.
Lecture 24
The Laplacian
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The Laplacian
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 introduce yet another important operator for our studies of electromagnetism.
Lecture 25
Curl of the Gradient
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Curl of the Gradient
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 what happens when we take the curl of the grad of a vector field.
Lecture 26
Divergence of the Curl
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Divergence of the Curl
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 what happens when we take the divergence of the curl of a vector field.
Lecture 27
Curl of the Curl
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Curl of the Curl
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 what happens when you take the double curl of a vector function.
Lecture 28
Fundamental Theorem of Calculus
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Fundamental Theorem of Calculus
In this video I continue with my tutorials which cover the necessary vector calculus for classical electromagnetism which is pitched at university undergraduate level.
Lecture 29
Fundamental Theorem for Gradients
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Fundamental Theorem for Gradients
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 extend the Fundamental Theorem of Calculus to three dimensions.
Lecture 30
Green's Divergence Theorem
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Green's Divergence Theorem
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 the fundamental theorem for divergences.
Lecture 31
Stokes' Theorem
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Stokes' Theorem
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 the fundamental theorem for curls.
Lecture 32
Integration by Parts Rule 1
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Integration by Parts Rule 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 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 33
Integration by Parts Example
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Integration by Parts Example
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 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 34
Integration by Parts Rule 2
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Integration by Parts Rule 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 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 35
Integration by Parts Rule 3
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Integration by Parts Rule 3
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 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 36
Integration by Parts Rule 4
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Integration by Parts Rule 4
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 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 37
Spherical Polar Co-ordinates
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Spherical Polar Co-ordinates
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 transforming from rectangular to spherical polar co-ordinates. This greatly simplifies calculations when there exists spherical symmetry.
Lecture 38
Helmholtz Theorem (No Derivation)
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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 39
Dirac Delta Function (1/2)
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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 40
Dirac Delta Function (2/2)
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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 41
Gradient of One Over the Separation Vector
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Gradient of One Over the Separation Vector
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 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 42
Laplacian of One Over the Separation Vector
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Laplacian of One Over the Separation Vector
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.
Lecture 43
Helmholtz Theorem Proof (Part 1)
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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 44
Helmholtz Theorem Proof (Part 2)
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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 45
Derivation Biot and Savart Law
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Derivation Biot and Savart Law
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 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.