Suppose that (x(t),y(t)), for (( a (less than or equal to) t (less than or equal

Question

Suppose that (x(t),y(t)), for (( a (less than or equal to) t (less than or equal to) b)), is a parameterization of the smooth curve C, with endpoints P = (x(a), y(a)) and Q = (x(b), y(b)). The values of the multivariable function f along C are then given by the single variable function h(t) = f(x(t),y(t)), with (( a (less than or equal to) t (less than or equal to) b)).

(a) Using the chain rule for multivariable functions, find h'(t) in terms of fx, fy, x', and y'.

(b) Use the Fundamental Theorem of Calculus applied to h(t) to show that (integral over C)

gradf

Explanation / Answer

Suppose that (x(t),y(t)), for a<t<b is a parameterization of the smooth curve C, with endpoints P=(x(a),y(a)) and Q=(x(b),y(b)). The values of the multivariable function f along C are then given by the single variable function h(t)=f(x(t),y(t)), with a<t<b.

a.Using the chain rule for multivariable functions, find h'(t) in terms of Fx,Fy,x' and y'.

b. Use the fundemental theorem of calculus applied to h(t) to show that
?c gradf (dot) dr= f(Q)-f(P)

a) dh/dt = ?f/?x dx/dt + ?f/?y dy/dt, by the Chain Rule.

b) ?c ?f

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.