Electromagnetic fields, lecture 2 - differential vector calculus in rectangular coordinates

Overview

Tim Gorringe introduces differential vector calculus in rectangular coordinates, defining the del operator and using it to derive divergence, curl, and gradient. He connects each operation to a physical quantity: divergence measures flux per unit volume, curl measures circulation per unit area, and the gradient describes scalar-field change along a step.

Key takeaways

In rectangular coordinates, divergence is ∂Aₓ/∂x + ∂Aᵧ/∂y + ∂A_z/∂z, and multiplying it by a small volume gives the net outward flux from that volume.
Divergence arises because opposing faces of a small box carry opposite outward area vectors; only changes in the field components across the box contribute to net flux.
The curl ∇×A is a vector whose dot product with an oriented area element gives local circulation around the boundary of that area.
The 3×3 determinant with coordinate unit vectors, partial derivatives, and field components is a compact method for organizing the rectangular-coordinate curl and its signs.
The gradient ∇u points according to the scalar field's spatial changes, and ∇u·ds gives the first-order change in u along a small displacement.

Chapters

0:00 Electromagnetic Fields and the Vector Calculus Course Plan

Electromagnetic fields are vector fields, with a vector value at every point in space; temperature provides an example of a scalar field.
The course will develop differential and integral vector calculus, starting with rectangular coordinates before adding cylindrical and spherical coordinates.
The operations introduced in this lecture help describe field variation, sources, and interactions.

3:00 The Del Operator and Its Three Field Operations

In rectangular coordinates, the del operator combines the unit vectors x̂, ŷ, and ẑ with partial derivatives ∂/∂x, ∂/∂y, and ∂/∂z.
Del is treated as a vector operator: it transforms like a vector under coordinate rotation and acts on a scalar or vector field.
The dot product ∇·A defines divergence, the cross product ∇×A defines curl, and ∇u defines the gradient of a scalar field.

9:00 Deriving Divergence in Rectangular Coordinates

Writing ∇ and a vector field A in components reduces divergence to a dot product.
Because rectangular unit vectors are orthonormal, only matching components survive: x̂·x̂, ŷ·ŷ, and ẑ·ẑ each equal 1.
The resulting formula is ∇·A = ∂Aₓ/∂x + ∂Aᵧ/∂y + ∂A_z/∂z, a scalar quantity.

13:00 Divergence as Net Flux per Unit Volume

For an infinitesimal rectangular volume, Gorringe calculates the flux through the opposing faces perpendicular to the x-axis.
The outward area vectors point in opposite directions, so the net x-face flux depends on the change in Aₓ across the cube, not its constant value.
Dividing by the small x-length and multiplying by the volume element dV gives the contribution (∂Aₓ/∂x)dV.
Adding the y- and z-face contributions yields total flux (∇·A)dV, so divergence measures net outward flux per unit volume.

25:00 Deriving the Curl Components from Cross Products

Curl is formed by taking the cross product of the rectangular-coordinate del operator with A.
Parallel unit-vector pairs such as x̂×x̂ vanish, while perpendicular pairs survive and produce the third coordinate direction.
The six surviving products combine into three component pairs, with signs determined by cross-product order, such as ŷ×ẑ = x̂ and ẑ×ŷ = −x̂.
Each curl component uses derivatives of the other two field components with respect to the other two coordinates.

32:00 The Curl Determinant and Local Circulation

A compact way to remember the rectangular-coordinate curl is a 3×3 determinant with unit vectors in the first row, partial derivatives in the second, and field components in the third.
Expanding the determinant reproduces the component pairs and their signs without memorizing each term separately.
The dot product of curl with an oriented area element represents circulation around the loop bounding that area.
Curl therefore measures local circulation, or rotation, per unit area, paralleling divergence's flux-per-volume interpretation.

38:00 The Scalar Gradient in Rectangular Coordinates

Applying ∇ to a scalar field u gives ∇u = x̂(∂u/∂x) + ŷ(∂u/∂y) + ẑ(∂u/∂z).
Unlike divergence, which produces a scalar, the gradient produces a vector containing the scalar field's directional rates of change.
The dot product ∇u·ds gives the incremental change in u for a small displacement ds.

42:00 Summary of Divergence, Curl, and Gradient

Divergence is a scalar formed from the sum of the three matching component derivatives and describes local flux per unit volume.
Curl is a vector that describes local circulation per unit area, while the 3×3 determinant organizes its rectangular-coordinate terms.
Gradient is a vector of partial derivatives, and its dot product with a displacement gives the corresponding change in the scalar field.

46:00 Next Topic: Line and Surface Integrals

The next class will introduce integral vector calculus, beginning with line integrals and surface integrals.
Gorringe plans to follow the definitions with worked illustrations and examples in the upcoming classes.

Transcript excerpt

0:01 Okay. Um, last class we talked a little bit about the administration of the class. Uh, if you got any questions on the administration of the class that were not clear in the syllabus, which is quite possible, then just obviously just let me know. We'll figure that out. Um, we then spoke a little bit about two topics. One was just introduced in electromagnetic field. So the concepts of electromagnetic fields, it was really a lot of works about electromagnetic fields. And then I did a really quick uh a really speedy run through of vector algebra at the end of the class because I'm assuming that you're comfortable with vector algebra and adding vectors and um uh the the products of vectors

0:52 specifically the cross productduct and the dot product uh the vector product and the scalar product because we'll be going to use use those in this this class a lot. Um so today's class we're going to start our work on um vector calculus and this work on vector calculus will cover I think uh including this class the next six classes. So uh we're going to be spending two two weeks on vector calculus. Today's class is going to be specific to differential vector calculus. The next class Friday will be an introduction to integral vector calculus. Uh in those two classes we're going to be doing integral and

1:40 differential vector calculus in rectangular coordinates. And then it's next week that we're going to introduce additional coordinate systems. We're going to introduce cylindrical coordinates. We're going to introduce spherical coordinates. So we're going to expand that sort of toolbox of integral differential vector calculus uh to make it capable or suitable to solving more problems that have rectangular or cylindrical or spherical symmetries. So that's that's

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In this course

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, tim gorringe.