MTHSC 206 Section 13.2 Derivtives nd Integrls of Vector Functions
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
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 ).
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 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)
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.
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.
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)).
Exmple Show tht if r(t) = c is constnt, then r (t) is orthogonl to r(t) for ll t.
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) )?
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
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
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.
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().
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).
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.