A note on the gradient of a function.

I want to insert a note on (gradient of ). 1. It is not perpendicular to everything in the surface. Most proofs only go as far as to prove it is perpendicular to continuous parameterized curves. Nothing more. Stop reading too deeply into it. 2. It is mostly useful for finding perpendiculars to all parameterizedContinue reading “A note on the gradient of a function.”

The chain rule in multi-variable calculus: Generalized

Now we’ll discuss the chain rule for -nested functions. For example, an -nested function would be . What would be? We know that . If is continuous, then such that , which is equivalent to saying . In turn such that . Hence, we have Continuing like this, we get the formula such that forContinue reading “The chain rule in multi-variable calculus: Generalized”

Multi-variable differentiation.

There are very many bad books on multivariable calculus. “A Second Course in Calculus” by Serge Lang is the rare good book in this area. Succinct, thorough, and rigorous. This is an attempt to re-create some of the more orgasmic portions of the book. In space, should differentiation be defined as ? No, as divisionContinue reading “Multi-variable differentiation.”