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Consider the following C: line segments from (0, 1) to (0, 4) and (0, 4) to (3, 3) (a) Find a parametrization of the path C. 3,53 r(t) (b) Evaluate (4x + 9y ds. Consider the following C: line se...
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of...
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
(1) Evaluate the following line integrals in R3. r +yds for C the line segment from (0, 1,0) to (1, 0,0) for C the line segment from (0,1,1) to (1,0,1). for C the circle (0, a cos t, a sin t) for O (...
(1) Evaluate the following line integrals in R3. r +yds for C the line segment from (0, 1,0) to (1, 0,0) for C the line segment from (0,1,1) to (1,0,1). for C the circle (0, a cos t, a sin t) for O (iv) 2π, with a a positive constant. t for C the curve (cost +tsint,sint tcost, 0) for Osts v3 (Hint for (i): use the parametrization (z, y, z) = (t, 1-t, 0) for 0 1) t
(1)...
Find a parametrization of the tangent line at the point indicated. r(t) = (1 - 4,...
Find a parametrization of the tangent line at the point indicated. r(t) = (1 - 4, 4t, 5t), t = 2
1. Evaluate the line integral S3x2yz ds, C: x = t, y = t?, z =...
1. Evaluate the line integral S3x2yz ds, C: x = t, y = t?, z = t3,0 st 51. 2. Evaluate the line integral Scyz dx - xz dy + xy dz , C: x = e', y = e3t, z = e-4,0 st 51. 3. Evaluate SF. dr if F(x,y) = x?i + xyj and r(t) = 2 costi + 2 sin tj, 0 st St. 4. Determine whether F(x,y) = xi + yj is a conservative vector field....
Evaluate the following line integral. ds; C is the line r(s) = for osss 4 72...
Evaluate the following line integral. ds; C is the line r(s) = for osss 4 72 72
7. (10 points) Consider the linear path "C" in the xy-plane from the point (1,1) to...
7. (10 points) Consider the linear path "C" in the xy-plane from the point (1,1) to the point (2,3). (a) Find an appropriate vector valued function r(t) along with an appropriate interval a <t<b which describes the path C. (b) Evaluate the line integral 2xy ds where C is the curve you found above.
QB(27pts)(a). Evaluate the circulation ofF(xy)-<x,y+x> on the curve r(t)=<2cost, 2sinp, foross2n (b) Evaluate J F.dr, where C is a piecewise smooth path from (1,0) to (2,1) and F- (e'...
QB(27pts)(a). Evaluate the circulation ofF(xy)-<x,y+x> on the curve r(t)=<2cost, 2sinp, foross2n (b) Evaluate J F.dr, where C is a piecewise smooth path from (1,0) to (2,1) and F- (e'cos x)i +(e'sinx)j [Hint: Test F for conservative (c). Use green theorem to express the line integral as a double integral and then evaluate. where C is the circle x+y-4 with counterclockwise orientation. (d(Bonus10 pts) Consider the vector field Foxyz) a. Find curl F y, ,z> F.dr where C is the curve...
(e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(...
Problem 4.9
(e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
find the average value of the following function Evaluate the following line integral along each path C--G is the line segment from (-5,-3) to (0,2). C Gis the arc ofthe parabola x-4-y2from (-5,-...
find the average value of the following function
Evaluate the following line integral along each path C--G is the line segment from (-5,-3) to (0,2). C Gis the arc ofthe parabola x-4-y2from (-5,-3)to (0,2) a. b. from (5,- 3) (12 points)|y'dx+ xdy
Evaluate the following line integral along each path C--G is the line segment from (-5,-3) to (0,2). C Gis the arc ofthe parabola x-4-y2from (-5,-3)to (0,2) a. b. from (5,- 3) (12 points)|y'dx+ xdy
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3), 0StS 4 Evaluate F. dr Justify...
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3), 0StS 4 Evaluate F. dr Justify your answer. iii. Find a function y: R3-+ R such that F iv. Evaluate F.dr where「is the path y =r', z = 0, from (0.0.0) to (2.8.0) followed by the line segment from (2,8,0) to (1,1,2) 22 marks)
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3),...
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
(1) Evaluate the following line integrals in R3. r +yds for C the line segment from (0, 1,0) to (1, 0,0) for C the line segment from (0,1,1) to (1,0,1). for C the circle (0, a cos t, a sin t) for O (iv) 2π, with a a positive constant. t for C the curve (cost +tsint,sint tcost, 0) for Osts v3 (Hint for (i): use the parametrization (z, y, z) = (t, 1-t, 0) for 0 1) t
(1)...
Find a parametrization of the tangent line at the point indicated. r(t) = (1 - 4, 4t, 5t), t = 2
1. Evaluate the line integral S3x2yz ds, C: x = t, y = t?, z = t3,0 st 51. 2. Evaluate the line integral Scyz dx - xz dy + xy dz , C: x = e', y = e3t, z = e-4,0 st 51. 3. Evaluate SF. dr if F(x,y) = x?i + xyj and r(t) = 2 costi + 2 sin tj, 0 st St. 4. Determine whether F(x,y) = xi + yj is a conservative vector field....
Evaluate the following line integral. ds; C is the line r(s) = for osss 4 72 72
7. (10 points) Consider the linear path "C" in the xy-plane from the point (1,1) to the point (2,3). (a) Find an appropriate vector valued function r(t) along with an appropriate interval a <t<b which describes the path C. (b) Evaluate the line integral 2xy ds where C is the curve you found above.
QB(27pts)(a). Evaluate the circulation ofF(xy)-<x,y+x> on the curve r(t)=<2cost, 2sinp, foross2n (b) Evaluate J F.dr, where C is a piecewise smooth path from (1,0) to (2,1) and F- (e'cos x)i +(e'sinx)j [Hint: Test F for conservative (c). Use green theorem to express the line integral as a double integral and then evaluate. where C is the circle x+y-4 with counterclockwise orientation. (d(Bonus10 pts) Consider the vector field Foxyz) a. Find curl F y, ,z> F.dr where C is the curve...
Problem 4.9
(e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
find the average value of the following function
Evaluate the following line integral along each path C--G is the line segment from (-5,-3) to (0,2). C Gis the arc ofthe parabola x-4-y2from (-5,-3)to (0,2) a. b. from (5,- 3) (12 points)|y'dx+ xdy
Evaluate the following line integral along each path C--G is the line segment from (-5,-3) to (0,2). C Gis the arc ofthe parabola x-4-y2from (-5,-3)to (0,2) a. b. from (5,- 3) (12 points)|y'dx+ xdy
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3), 0StS 4 Evaluate F. dr Justify your answer. iii. Find a function y: R3-+ R such that F iv. Evaluate F.dr where「is the path y =r', z = 0, from (0.0.0) to (2.8.0) followed by the line segment from (2,8,0) to (1,1,2) 22 marks)
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3),...
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