Multivariable Chain Rule, Real Analysis II

In this video, we explore the multivariable chain rule, starting with the question of why matrix multiplication is defined the way it is. By understanding matrix multiplication as the composition of linear transformations, we arrive at the chain rule in multivariable calculus: if f is differentiable at p and g is differentiable at f(p), then g(f(x)) is differentiable at p, and D(gof)(p) is Dg(f(p))Df(p). In other words, the Jacobian matrix of a composition of functions is given by the product (composition) of their individual Jacobians. (Proof here: https://youtu.be/mnPqoTmOV4w) Moving to examples, we look at a common form of the chain rule in cases where functions depend on layered inputs, highlighting how the chain rule produces familiar formulas. In our second example, we apply the chain rule in the context of a scalar function evaluated along a curve, translating the chain rule into a dot product involving the gradient of the function and the velocity vector of the curve. This approach is efficient, saving us from directly composing functions. #realanalysis #advancedcalculus #ChainRule #MatrixMultiplication #Jacobian #MultivariableCalculus #Gradients #Differentiability #VectorCalculus #calculus3