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Multivariable Functions
How different is multivariable calculus from its single-variable counterpart? When I was a student, I had a professor who used to say something like, “multivariable and single-variable functions behave the same, you just have to write more.”
Well, this couldn’t be further from the truth. Just think about what we are doing in machine learning: training models with gradient descent; that is, finding a configuration of parameters that minimize a parametric function. In one variable (which is not a realistic assumption), we can do this with the derivative, as we saw in Section 13.2. How can we extend the derivative to multiple dimensions?
The inputs of multivariable functions are vectors. Thus, given a function f : ℝn →ℝ, we can’t just define
to the analogue of Definition 54. Why? Because the division with the vector x0 −x is not defined.
As we’ll see, differentiation in multiple dimensions...
