Kevin James. MTHSC 206 Section 13.2 Derivatives and Integrals of Vector

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1 MTHSC 206 Section 13.2 Derivtives nd Integrls of Vector Functions

2 Definition Suppose tht r(t) is vector function. We define its derivtive by [ ] dr r(t + h) r(t) dt = r (t) = lim h 0 h

3 Definition Suppose tht r(t) is vector function. We define its derivtive by [ ] dr r(t + h) r(t) dt = r (t) = lim h 0 h Note When r (t 0 ) exists nd is nonzero, we refer to it s the tngent vector to the spce curve defined by r(t) t the point P = r(t 0 ).

4 Definition Suppose tht r(t) is vector function. We define its derivtive by [ ] dr r(t + h) r(t) dt = r (t) = lim h 0 h Note When r (t 0 ) exists nd is nonzero, we refer to it s the tngent vector to the spce curve defined by r(t) t the point P = r(t 0 ). In this cse, the line tngent to the curve t P = r(t 0 ) is given by P + r (t 0 )t.

5 Definition Suppose tht r(t) is vector function. We define its derivtive by [ ] dr r(t + h) r(t) dt = r (t) = lim h 0 h Note When r (t 0 ) exists nd is nonzero, we refer to it s the tngent vector to the spce curve defined by r(t) t the point P = r(t 0 ). In this cse, the line tngent to the curve t P = r(t 0 ) is given by P + r (t 0 )t. Definition The unit tngent vector to the curve is defined by T (t) = r (t) r (t)

6 Theorem Suppose tht r(t) = (f (t), g(t), h(t)) = f (t)i + g(t)j + h(t)k, where f, g nd h re differentible functions. Then, r (t) = (f (t), g (t), h (t)) = f (t)i + g (t)j + h (t)k.

7 Theorem Suppose tht r(t) = (f (t), g(t), h(t)) = f (t)i + g(t)j + h(t)k, where f, g nd h re differentible functions. Then, r (t) = (f (t), g (t), h (t)) = f (t)i + g (t)j + h (t)k. Exmple Find the derivtive of r(t) = (cos(t), sin(t), 2t). Find the unit tngent vector t the point where t = π.

8 Theorem (Properties of Derivtives) Suppose tht u, v re differentible vector functions, c R nd f (t) is rel vlued function. Then, 1 d dt [u(t) + v(t)] = u (t) + v (t). 2 d dt [cu(t)] = cu (t). 3 d dt [f (t)u(t)] = f (t)u(t) + f (t)u (t). 4 d dt [u(t) v(t)] = u (t) v(t) + u(t) v (t). 5 d dt [u(t) v(t)] = u (t) v(t) + u(t) v (t). 6 d dt [u(f (t))] = f (t)u (f (t)).

9 Exmple Show tht if r(t) = c is constnt, then r (t) is orthogonl to r(t) for ll t.

10 Exmple Show tht if r(t) = c is constnt, then r (t) is orthogonl to r(t) for ll t. Exmple Wht is the derivtive of r(t) = (cos(sin(t)), sin(sin(t)), e sin(t) )?

11 Integrtion Definition We define the definite integrl of continuous vector function r(t) = (f (t), g(t), h(t)) on n intervl [, b] s b r(t)dt = lim r(t i ) t

12 Integrtion Definition We define the definite integrl of continuous vector function r(t) = (f (t), g(t), h(t)) on n intervl [, b] s b = r(t)dt = lim ( lim r(t i ) t f (t i ) t, lim g(t i ) t, lim ) h(t i ) t

13 Integrtion Definition We define the definite integrl of continuous vector function r(t) = (f (t), g(t), h(t)) on n intervl [, b] s b = r(t)dt = lim ( lim r(t i ) t f (t i ) t, lim g(t i ) t, lim ) h(t i ) t Fct If r(t) = (f (t), g(t), h(t), then b ( b r(t)dt = b b ) f (t)dt, g(t)dt, h(t)dt.

14 Theorem (Fundmentl Theorem of Clculus) Suppose tht R(t) nd r(t) re continuous vector vlued functions with R (t) = r(t). Then, b r(t)dt = [R(t)] b = R(b) R().

15 Theorem (Fundmentl Theorem of Clculus) Suppose tht R(t) nd r(t) re continuous vector vlued functions with R (t) = r(t). Then, b r(t)dt = [R(t)] b = R(b) R(). Note If R nd r re s bove, then we will use the nottion R(t) = r(t)dt for the indefinite integrl of r(t).

16 Theorem (Fundmentl Theorem of Clculus) Suppose tht R(t) nd r(t) re continuous vector vlued functions with R (t) = r(t). Then, b r(t)dt = [R(t)] b = R(b) R(). Note If R nd r re s bove, then we will use the nottion R(t) = r(t)dt for the indefinite integrl of r(t). Exmple Compute π/2 0 (cos(t), sin(t), 2t)dt.

Section Vector Functions and Space Curves

Section Vector Functions and Space Curves Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of