In Exercises 9-18, evaluate
F(x, y, z) =
C: Smooth curve from (0, 0, 0) to
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- Suppose F(x, y) = x² + y² and C is the line segment from point A = (1, −1) to B = (3, −5). (a) Find a vector parametric equation (t) for the line segment C so that points A and B correspond to t= 0 and t = 1, respectively. F(t) = = (b) Using the parametrization in part (a), the line integral of along C' is dt LE · dF = [ ° F (F(t)) - 7"' (t) dt = √° with limits of integration a = and b = (c) Evaluate the line integral in part (b).arrow_forwardDoes f(z)=z/(sin z)^2 have a pole of order 1 or 2 at z=0?arrow_forwardEvaluate the line integral (ci+ 3xyj – (x + z)k) · dr where C is the parametric curve r(t) = (1 – t)i + (4 + t)j + (2 – t)k, 0arrow_forward3. Let F = (2y² + z)i + 4.ryj + «k. (a) Show whether F is conservative. (b) Using the fundamental theorem of calculus for line integrals, prove that F. dr = -T along a curve C defined by a vector function r(t) = cos ti + sin tj + tk for 0arrow_forwardF(1, y.:)= (x², 2x²y*, 5yz*) XZ Given: Find (a) div F (b) curl Farrow_forwardI need help with this problem and an explanation for the solution described below (Vector-Valued Functions, Derivatives and integrals, Vector fields)arrow_forwarda) Show that F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j is the gradient of some function f. Find f b) Evaluate the line integral ʃC F dr where the vector field is given by F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j and C is the curve on the circle x 2 + y 2 = 9 from (3, 0) to (0, 3) in a counterclockwise direction.arrow_forwardEvaluate the line integral ∫CF→⋅dr→ using the Fundamental Theorem of Line Integrals if F→(x,y)=(4x+4y)i→+(4x+4y)j→and Cis the smooth curve from (−1,1)to (5,6). Enter the exact answer. ∫CF→⋅dr→=arrow_forwardSketch and describe the vector field F (x, y) = (-y,2x)arrow_forwardarrow_back_iosarrow_forward_iosRecommended textbooks for you
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