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1 point) The path C is a line segment of length 39 in the plane starting at (2,1). For f(z, y)- 5x 12y, consider (a) Wh...
9. (10pts) Answer true or false: (a) The domain of f(x,y) = In(1-z?-уг) + Vi-z?-уг is the unit ball {(z, y): x2 + y2 1} . (b) The direction of the maximum rate of increase of g(x, y, 2yz at the point...
9. (10pts) Answer true or false: (a) The domain of f(x,y) = In(1-z?-уг) + Vi-z?-уг is the unit ball {(z, y): x2 + y2 1} . (b) The direction of the maximum rate of increase of g(x, y, 2yz at the point (1,1,1) is 2,1,1 (c) For F2y,2r3y1>, F-dr is independent of path in the plane. (d) × (▽ . F) makes sense. (e) ▽f.dr =4 where f(x, y, z) = zyz and C is the line segment starting at...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment starting at (1,0) and ending at (3, 1). Question 23 1 pts Consider the vector field F= (1, y). Compute the path integral of this field along the path: start at (0,0) and go up 2 units, then go right 3 units, then go down 4 units and stop. Question 24 1 pts Compute Ss(-y+ye*y)dx + (x + xey)dy, where S is the path:...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment starting at (1,0) and ending at (3,1).
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...
The path integral of a function f(x, y) along a path e in the xy-plane with...
The path integral of a function f(x, y) along a path e in the xy-plane with respect to a parameter r is given by 2. fex,y)ds= f(x),ye) /x(mF +y(t" dr , where a sr sb. (a) Show that the path integral of f(x, y) along a path c(0) in polar coordinates where r=r(0), α<θ<β, is Sf(r cos 0,rsin e) oN+( de. (b) Use this formula to compute the arc length of the path r 1+cos0, 0<0 27
The path integral...
true or false is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If...
true or false
is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Det...
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Determine the maximum value of Daf(P), where u is any unit vector at P c) Find the angle between Vfp and PO, where O denotes the origin. d) Find an equation for the tangent plane to S at P
rty. I 5. [16...
2. Consider the line segment & the zz-plane given by cone C by rotating the line about the z-axis...
2. Consider the line segment & the zz-plane given by cone C by rotating the line about the z-axis. fromz 0 to z4. We can then obtain a 2 (a) 4 pts Find a parametric representation r(u, v) for C, including bounds for u and v. (b) (4 pts Calculate and simplify r x rl (c) 3 pts Use a double integral to find the surface area of C
2. Consider the line segment & the zz-plane given by cone...
2-1/2dz if C is a polygonal line with vertices 2,1 + i,-1 i,-2 (without the segment [-2,2) and z-...
2-1/2dz if C is a polygonal line with vertices 2,1 + i,-1 i,-2 (without the segment [-2,2) and z-1/ is a principal value. Hint: consider a particular branch which is analytic on the contour uate the following integrals (all contours are positively oriented): cosh(z) 3 dz if C is a square of vertices 1 ti,-1ti C 2 sin(2) dz if C is a circle 3 2(2,2 2 3 dz if C is a rectangle with sides along the lines x-1,x--1,y...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z +...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
9. (10pts) Answer true or false: (a) The domain of f(x,y) = In(1-z?-уг) + Vi-z?-уг is the unit ball {(z, y): x2 + y2 1} . (b) The direction of the maximum rate of increase of g(x, y, 2yz at the point (1,1,1) is 2,1,1 (c) For F2y,2r3y1>, F-dr is independent of path in the plane. (d) × (▽ . F) makes sense. (e) ▽f.dr =4 where f(x, y, z) = zyz and C is the line segment starting at...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment starting at (1,0) and ending at (3, 1). Question 23 1 pts Consider the vector field F= (1, y). Compute the path integral of this field along the path: start at (0,0) and go up 2 units, then go right 3 units, then go down 4 units and stop. Question 24 1 pts Compute Ss(-y+ye*y)dx + (x + xey)dy, where S is the path:...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment starting at (1,0) and ending at (3,1).
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...
The path integral of a function f(x, y) along a path e in the xy-plane with respect to a parameter r is given by 2. fex,y)ds= f(x),ye) /x(mF +y(t" dr , where a sr sb. (a) Show that the path integral of f(x, y) along a path c(0) in polar coordinates where r=r(0), α<θ<β, is Sf(r cos 0,rsin e) oN+( de. (b) Use this formula to compute the arc length of the path r 1+cos0, 0<0 27
The path integral...
true or false
is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Determine the maximum value of Daf(P), where u is any unit vector at P c) Find the angle between Vfp and PO, where O denotes the origin. d) Find an equation for the tangent plane to S at P
rty. I 5. [16...
2. Consider the line segment & the zz-plane given by cone C by rotating the line about the z-axis. fromz 0 to z4. We can then obtain a 2 (a) 4 pts Find a parametric representation r(u, v) for C, including bounds for u and v. (b) (4 pts Calculate and simplify r x rl (c) 3 pts Use a double integral to find the surface area of C
2. Consider the line segment & the zz-plane given by cone...
2-1/2dz if C is a polygonal line with vertices 2,1 + i,-1 i,-2 (without the segment [-2,2) and z-1/ is a principal value. Hint: consider a particular branch which is analytic on the contour uate the following integrals (all contours are positively oriented): cosh(z) 3 dz if C is a square of vertices 1 ti,-1ti C 2 sin(2) dz if C is a circle 3 2(2,2 2 3 dz if C is a rectangle with sides along the lines x-1,x--1,y...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
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