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13.2
Derivatives and Integrals of Vector Functions
Derivatives We can build derivatives of vector functions from derivative of components, but the definition can also be done from first principles, with difference quotients: Definition (Derivative of r). The derivative r of a vector function of variable t is given by dr r(t + h) − r(t) = r = lim . h→0 dt h (1)
It can be checked that for r(t) = f (t), g(t), h(t) , the derivative (if it exists) is the vector of derivatives of the components: dr d = f (t), g(t), h(t) = f (t), g (t), h (t) dt dt Thus the derivative exists if and only if all of the component derivatives exists. Tangent Vectors For any value of t, where the derivative vector r (t) exists and is non-zero, it is tangent to the space curve C at point P with position vector r(t), and so is called the tangent vector to C at P . The line through P with this tangent vector is the tangent line to C at P . It will often be useful to consider the unit tangent vector T (t) = See Examples 1,2,3. Higher Derivatives and Smooth Curves Second and higher order derivatives are defined in the natural way. For example the second derivative of vector function r is the derivative of C, which is (r ) , but more compactly denoted r . Definition. A space curve is smooth if it is given by r(t) on interval I with both r and r continuous, and with r = 0 except possibly at the endpoints of I. If the derivative is zero at a finite number of points, the curve is piecewise smooth. Exercise. Explain why the condition r = 0 is important for smoothness. Example 4 should help. Differentiation Rules The familiar differentiation rules for sums, products and compositions have natural counterparts for vector functions: r (t) |r (t)| (2)
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CHAPTER 13. VECTOR FUNCTIONS
Theorem. For u and v differentiable vector functions, f a differentiable scalar (real-valued) function, and c a scalar constant, 1. 2. 3. 4. 5. 6. d [u(t) + v(t)] = u (t) + v (t). dt d [cu(t)] = cu (t). dt d [f (t)u(t)] = f (t)u(t) + f (t)u (t). dt d [u(t) · v(t)] = u (t) · v(t) + u(t) · v (t). dt d [u(t) × v(t)] = u (t) × v(t) + u(t) × v (t). dt d [u(f (t))] = u (f (t))f (t). dt
As always with the cross product, the order matters in item 5. See Example 5. Definite Integrals Like derivatives, definite integrals of vector functions can be built from first principles with Riemann sums, and one gets the predictable result in terms of integrals of components: For r(t) = f (t), g(t), h(t) ,
b a
r(t) dt =
a
b
f (t) dt,
a
b
b
g(t) dt,
a
h(t) dt
The Fundamental Theorem of Calculus The following should be no surprise: Theorem (The Fundamental Theorem of Calculus for Vector Valued Functions). If F , G and H are any anti-derivatives of f , g and h respectively, then R(t) = F (t), G(t), H(t) is an antiderivative of r; that is, R = r. Also
b a
r(t) dt = R(b) − R(a) = R(t)
b a
.
Other integral rules can be extended, but it is best to just to deal with such integration problems as a collection of three separate integrals. See Example 6. Homework Exercises 3, 4, 5*, 7, 9-14, 17-21, 22*, 23, 24*, 25, 26 (differentiation); 33-40 (integration).