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MathTheBeautiful Tensors vs Space-time Numbers 2: Gradient of a Scalar Function

331 views · Published 1 June 2017 · 16:44 · Indexed 18 October 2026

Channel: Doug Sweetser · 2017 · Science & Technology

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MathTheBeautiful (Pavel Grinfeld) has a series of tensors. Tensors a great, but I am hoping that space-time numbers - a technical variation on quaternions - might be better for mathematical physics.

This video is a reply to https://www.youtube.com/watch?v=p75-f0gN5c0&t=371s where Pavel Grinfeld explains on gradients in stretched coordinates.

In this video, the gradient of a scalar function in stretched coordinates is analyzed. To make sure the definition of the gradient of a scalar function does not change going to stretched coordinates, one must use a basis vector such as i/|i|^2.

With space-time numbers, notice that the gradient operator will change in the stretched coordinates. That nicely cancels what happens with the derivative.

The video finishes by looking at what happens if all the places for operators and functions are filled in. If so, key elements of electromagnetism just show up, even though we were just doing math.

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