DIRECTIONAL DERIVATIVES ALONG CURVES. Suppose that r(t) is a parametric curve that passes through the point (a,b) when t = 0. The average rate of change of a function f(x,y) along r(t) from t=0 to t=T is given by f(r(T)) f(r(0)) T If we take the limit as T +0 we get the instantaneous rate of change of f along r(t) at the point where t = 0, i.e., at (a, b). Let's write Drf(a,b) for this limit (assuming it exists) and call it the directional derivative of f along the curve r(t) at (a,b). d (a) Show that Drf(a, b) = f(r(t)). dt It=0 (b) If r(t) is a straight line with unit direction vector u that passes through (a,b) at t = 0, show that Drf(a,b) = Duf(a, b) is the usual directional derivative. (c) If f is differentiable at (a,b), use (a) and the chain rule to show that Drf(a,b) = Vf(a,b) r'(t). (d) Show that (c) reduces to the usual formula Duf(a, b) = Vf(a,b). u in case r(t) is a line as in (b). DIRECTIONAL DERIVATIVES ALONG CURVES. Suppose that r(t) is a parametric curve that passes through the point (a,b) when t = 0. The average rate of change of a function f(x,y) along r(t) from t=0 to t=T is given by f(r(T)) f(r(0)) T If we take the limit as T +0 we get the instantaneous rate of change of f along r(t) at the point where t = 0, i.e., at (a, b). Let's write Drf(a,b) for this limit (assuming it exists) and call it the directional derivative of f along the curve r(t) at (a,b). d (a) Show that Drf(a, b) = f(r(t)). dt It=0 (b) If r(t) is a straight line with unit direction vector u that passes through (a,b) at t = 0, show that Drf(a,b) = Duf(a, b) is the usual directional derivative. (c) If f is differentiable at (a,b), use (a) and the chain rule to show that Drf(a,b) = Vf(a,b) r'(t). (d) Show that (c) reduces to the usual formula Duf(a, b) = Vf(a,b). u in case r(t) is a line as in (b)