13.2: Calculus of Vector-Valued Functions

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

332

Pages:

2
Author:

Alexis H.

13.2 Calculus of Vector-Valued Functions This is straightforward because the techniques of single-variable calculus carry over with little change. What is new and important, however, is the geometric interpretation of the derivative as a tangent vector Definition: Limit of a vector-valued function A vector-valued function r(t) approaches the limit u (a vector) as t approaches t(0). In this case we write… lim r (t) = u t!t0 We can visualize the limit of a vector-valued function as a vector r(t) moving toward the limit vector u Theorem 1: Vector-valued limits are computed component-wise A vector-valued function r(t) = approaches a limit as t —> t(0) is and only if each component approaches a limit, and in this case, lim r (t) = hlim x (t) ; lim y (t) ; lim z (t)i z ret t!t0 y t!t0 t!t0 t!t0 We define the derivative of r(t) as the limit of the difference quotient: a r0 (t) = d r (t + h) ¡ r (t) r (t) = lim h!0 dt h We say that r(t) is differentiable at t if the limit in the equation above exists, and we say that r is differentiable if it is differentiable at all t in its domain Theorem 2: Vector-valued derivatives are computed component-wise A vector-valued function r(t) = is differentiable if and only if each component is differentiable. In this case, r0 (t) = d r(t) = hx0 (t) ; y 0 (t) ; z 0 (t)i dt Differentiation rules Assume that r(t), r1(t), and r2(t) are differentiable. Then 0 • Sum rule: (r1 (t) + r2 (t)) = r10 (t) + r20 (t) 0 (cr (t)) = cr0 (t) • Constant multiple rule: for any constant c, • Scalar product rule: for any differentiable scalar-valued function f, • Chain rule: for any differentiable scalar-valued function g, d (f (t) r (t)) = f 0 (t) r (t) + f (t) r0 (t) dt d r (g (t)) = r0 (g (t)) g 0 (t) dt Theorem 3: Product rules for dot and cross products Dot product rule: d (r1 (t) ¢ r2 (t)) = r10 (t) ¢ r2 (t) + r1 (t) ¢ r20 (t) dt Cross product rule: d (r1 (t) £ r2 (t)) = (r10 (t) £ r2 (t)) + (r1 (t) £ r20 (t)) dt We have seen three product rules involving vector-valued functions: the scalar product rule, the dot product rule, and the cross product rule. Each has the same form as the single-variable product rule: the derivative of the first term "times" the second, plus the first term "times" the derivative of the second The derivative as a tangent vector The tangent vector (if it exists and is nonzero) is a direction vector for the tangent line to the curve. The tangent line then is the line with vector parametrization: L (t) = r (t0) + tr0 (t0) Vector-valued integration Z b r (t) dt = h a Z b x (t) dt; a Z b y (t) dt; a Z b z (t) dti a In the single-variable case, two functions with the same derivative differ by a constant. Similarly, two vector-valued functions with the same derivative differ by a constant vector (i.e., a vector that does not depend on t)