partial derivative (revisited)

You met the partial derivative in Volume I as the slope you get by freezing all variables but one and differentiating as usual. Revisiting it from a deeper vantage: the partial derivative of f with respect to x, written partial f / partial x, is the rate of change of f as you move strictly parallel to the x-axis, with every other coordinate held fixed. It is one number out of many — each input direction has its own partial.

Formally, partial f / partial x at (a, b) is the limit of (f(a + h, b) - f(a, b)) / h as h -> 0, which is just the ordinary one-variable derivative of the slice g(x) = f(x, b). That slicing is the whole trick and also the whole danger. Because each partial only probes one axis-parallel direction, the existence of all partials at a point tells you almost nothing about the function's behavior along slanted directions. A function can have both partials equal to zero at a point and still be wildly discontinuous there. This is the great gap that the total derivative and differentiability were invented to close.

Despite that warning, partials are the indispensable raw material of multivariable calculus. They assemble into the gradient (the row of all first partials), into the Jacobian matrix for vector outputs, and into the Hessian for second-order behavior. Mixed second partials — partial^2 f / (partial x partial y) versus partial^2 f / (partial y partial x) — are equal whenever they are continuous (Clairaut's / Schwarz's theorem), which is why order of differentiation usually does not matter in practice. In physics they are everywhere: pressure as a partial of energy, the components of the gradient in del, the diffusion term partial u / partial t in the heat equation.