Multivariable Calculus

Directional derivatives and differentiability

A directional derivative measures how fast f changes at a point when we move in a fixed direction. For a unit vector u, the directional derivative at a is D_uf(a)=\lim_{t\to 0}\frac{f(a+tu)-f(a)}{t}, whenever this limit exists. The partial derivatives are the special cases u=e_1 and u=e_2. A function may possess every directional derivative at a point and still fail to be differentiable there, so existence of directional derivatives alone does not guarantee differentiability. Differentiability at a means that there is a linear map L with f(a+h)-f(a)-L(h)=o(\|h\|). If f is differentiable, then L(h)=\nabla f(a)\cdot h and D_uf(a)=\nabla f(a)\cdot u, which is linear in u. Hence, if the directional derivatives exist but do not depend linearly on u, the function cannot be differentiable at a. Continuity is a separate property. Differentiability implies continuity, but continuity does not imply differentiability. Even when both partial derivatives exist and are zero, the quotient \frac{f(h,k)}{\sqrt{h^2+k^2}} must tend to 0 along every path to the origin, and a failure along a curve such as a parabola shows non-differentiability. A useful test is to compute the one-sided limits of f(tu)/t. If these one-sided limits differ, the directional derivative in direction u does not exist. For example, for f(x,y)=\frac{xy^2}{x^2+y^2} with f(0,0)=0, both partial derivatives at the origin vanish, but D_uf(0)=\frac{ab^2}{a^2+b^2} for u=(a,b), which is not linear in u, so f is not differentiable at the origin. The rate of change in the direction from a point P to a point Q is \nabla f(P)\cdot\hat u, where \hat u=\frac{Q-P}{\|Q-P\|} is the unit vector. For f=xy+z^2 at (1,2,0) moving toward (3,2,4), the gradient is (2,1,0) and \hat u=\frac{1}{\sqrt5}(1,0,2), so the rate is \frac{2}{\sqrt5}. When a direction is tangent to a curve defined as the intersection of two surfaces, the tangent is parallel to \nabla F_1\times\nabla F_2, and after normalizing this vector one computes \nabla f\cdot\hat a. Only the absolute value is asked in such problems, so the sign of the chosen direction does not matter.

Local extrema and saddle points

A critical point of a differentiable function f(x,y) is a point where \nabla f=0. To classify it, form the Hessian determinant D=f_{xx}f_{yy}-f_{xy}^2 at that point. If D>0 and f_{xx}>0, the point is a local minimum; if D>0 and f_{xx}<0, it is a local maximum; if D<0, it is a saddle point; and if D=0, the test is inconclusive. For instance, f=x^2-y^2 has D=-4<0 at the origin, so the origin is a saddle. A saddle point is a critical point where f takes both larger and smaller values in every neighbourhood. This matters when D=0, because then the second-derivative test gives no information. In such cases one restricts f to suitable curves through the point and checks the sign of f along each curve. If the signs differ arbitrarily close to the point, the point is a saddle. When f is defined only off the coordinate axes, the critical point equations may have several branches, and the sign of f_{xx} together with D must be checked on each branch before deciding which is a local minimum. A constrained extremum on a line can be found by substitution or by geometry. Maximizing 49-x^2-y^2 on a line ax+by=c is equivalent to minimizing x^2+y^2, and the minimum of x^2+y^2 is the square of the distance from the origin to the line, namely \frac{c^2}{a^2+b^2}. For a quadratic function with no linear term, the origin is a critical point, and its nature is determined by the eigenvalues of the symmetric matrix of the quadratic form. For 2\times 2 matrices, \mathrm{Trace}(A^2)=a^2+2bc+d^2, and the Hessian of this function has eigenvalues of both signs, so the zero matrix is a saddle point. To find a global maximum on an unbounded closed region, one compares the values at interior critical points, the values on each boundary edge (where each edge is a one-variable problem), and the limiting behaviour at infinity. For f=(x^2+y^2)e^{2-x-y} on the first quadrant, the exponential decay dominates the polynomial growth, so f\to 0 at infinity, and the maximum is found among the candidate points.