Labware - MA35 Multivariable Calculus - Three Variable Calculus



Chain Rule


If (x(t),y(t),z(t)) is a differentiable curve in the domain of a differentiable function f, then w(t) = f(x(t),y(t),z(t)) is a differentiable function of t and w'(t) = fx(x(t),y(t),z(t))x'(t) + fy(x(t),y(t),z(t))y'(t) + fz(x(t),y(t),z(t))z'(t).

The tangent line to the graph f(x(t0),y(t0),z(t0)) will then be (x(t0),y(t0),z(t0) + u(x'(t0),y'(t0),z'(t0)). The gradient vector of f at (x0,y0,z0) is defined to be ∇x(t0),y(t0),z(t0) = (fx(x0,y0,z0),fy(x0,y0,z0),fz(x0,y0,z0).

Note that the gradient vector is a vector in the domain of the function. The chain rule can then be written in vector notation as w'(t) = ∇f(x(t),y(t),z(t))⋅(x'(t),y'(t),z'(t)). It follows that the function f(x(t),y(t),z(t)) has a maximum or minimum when the gradient of f at (x(t),y(t)),z(t)) is perpendicular to the velocity vector (x'(t),y'(t),z'(t)) at that point. In particular, if w(t) = c, a constant, then w'(t) = 0 for all t and the velocity vector is perpendicular to the gradient vector at all points.



  • 1. Give a formula for the chain rule for the special case that y(t) and z(t) are constant functions. Do the same for the case that x(t) and z(t) are constant functions and then for the case that x(t) and y(t) are constant functions.
  • 2. Use the chain rule to find the maxima and minima for the function f(x, y, z) = x + y + z along the curve (cos(t), sin(t), cos(2t)).