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is this 32pi or 16pi? for 0 4, oriented inward (i.e. normal points 4y2422- 1. (15 pts) Let C be the cone toward the...
for 0r 4, oriented inward (i.e. normal points 4y2+422- 1. (15 pts) Let C be the cone toward the positive r-axis)....
for 0r 4, oriented inward (i.e. normal points 4y2+422- 1. (15 pts) Let C be the cone toward the positive r-axis). Use Stokes' theorem to evaluate Vx(0,1+y)- ds. Make sure to indicate how you are orienting the boundary
for 0r 4, oriented inward (i.e. normal points 4y2+422- 1. (15 pts) Let C be the cone toward the positive r-axis). Use Stokes' theorem to evaluate Vx(0,1+y)- ds. Make sure to indicate how you are orienting the boundary
Consider the unit cube with vertices (corner points) (0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1), (...
Consider the unit cube with vertices (corner points) (0, 0, 0),
(0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1),
(1, 1, 1). Let S be the boundary of the cube minus (i.e. not
including) the bottom square (the side which lies in the xy plane).
Orient S with the normal which points out of the cube. Let F =
<− y , x , y^2e^x . Evaluate (curl F) ·...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2,...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in the counterclockwise direction n is the outward-...
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in the counterclockwise direction n is the outward-pointing normal vector on , and C is the boundary (b) (15 points) Evaluate directly the line integral p F- nds in part (a).
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in...
4. -15 points Use Green's theorem for flux to evaluate the line integra ds . (6ху, y2-x2) and C i...
4. -15 points Use Green's theorem for flux to evaluate the line integra ds . (6ху, y2-x2) and C is the positively oriented boundary curve of the region bounded by y F 0 and y x(4-x). Submit Answer
4. -15 points Use Green's theorem for flux to evaluate the line integra ds . (6ху, y2-x2) and C is the positively oriented boundary curve of the region bounded by y F 0 and y x(4-x). Submit Answer
(2) Let F zi + xj+yk and consider the integral vx Fi n dS where S is the surface of the paraboloid z = 1-x2-y2 corr...
(2) Let F zi + xj+yk and consider the integral vx Fi n dS where S is the surface of the paraboloid z = 1-x2-y2 corresponding to 0, and n is a unit normal vector to S in the positive z-direction. (a) Apply Stokes' theorem to evaluate the integral. b) Evaluate the integral directly over the surface S. (c) Evaluate the integral directly over the new surface S which is given by the disk
(2) Let F zi + xj+yk...
Use Stokes' Theorem to evaluate S (double integral) curl F · dS. F(x, y, z) = x^2*y^3*z i + sin(xyz) j + xyz k, S is the part of the cone y^2 = x^2 + z^2 that lies between the planes y = 0 and y =...
Use Stokes' Theorem to evaluate S (double integral) curl F · dS. F(x, y, z) = x^2*y^3*z i + sin(xyz) j + xyz k, S is the part of the cone y^2 = x^2 + z^2 that lies between the planes y = 0 and y = 3, oriented in the direction of the positive y-axis.
F-dS where S is the cylinder x? +-2, 0 s y < 2 oriented by the unit normal 5- Let F(x,y,z)= (-6x,0,-62). Evaluat...
F-dS where S is the cylinder x? +-2, 0 s y < 2 oriented by the unit normal 5- Let F(x,y,z)= (-6x,0,-62). Evaluate pointing out of the cylinder. 6-Let F(x, y,2)- yi- xj +zx°y?k. Evaluate (Vx F) . dS where S is the surface x2+y+32 - 1, z <0 oriented by the upward- pointing unit normal.
F-dS where S is the cylinder x? +-2, 0 s y
Q2 13 Points Begin with the paraboloid = 22 + y2, for 0 < < 4,...
Q2 13 Points Begin with the paraboloid = 22 + y2, for 0 < < 4, and slice it with the plane y 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2x + y, 2x + z, 2y + x>. ZA с S 2 =...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outwa...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outward pointing normal in dS terms r and θ. This cone has an open bottom. . The integrand must be fully simplified » Do not evaluate the integral
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the...
for 0r 4, oriented inward (i.e. normal points 4y2+422- 1. (15 pts) Let C be the cone toward the positive r-axis). Use Stokes' theorem to evaluate Vx(0,1+y)- ds. Make sure to indicate how you are orienting the boundary
for 0r 4, oriented inward (i.e. normal points 4y2+422- 1. (15 pts) Let C be the cone toward the positive r-axis). Use Stokes' theorem to evaluate Vx(0,1+y)- ds. Make sure to indicate how you are orienting the boundary
Consider the unit cube with vertices (corner points) (0, 0, 0),
(0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1),
(1, 1, 1). Let S be the boundary of the cube minus (i.e. not
including) the bottom square (the side which lies in the xy plane).
Orient S with the normal which points out of the cube. Let F =
<− y , x , y^2e^x . Evaluate (curl F) ·...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in the counterclockwise direction n is the outward-pointing normal vector on , and C is the boundary (b) (15 points) Evaluate directly the line integral p F- nds in part (a).
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in...
4. -15 points Use Green's theorem for flux to evaluate the line integra ds . (6ху, y2-x2) and C is the positively oriented boundary curve of the region bounded by y F 0 and y x(4-x). Submit Answer
4. -15 points Use Green's theorem for flux to evaluate the line integra ds . (6ху, y2-x2) and C is the positively oriented boundary curve of the region bounded by y F 0 and y x(4-x). Submit Answer
(2) Let F zi + xj+yk and consider the integral vx Fi n dS where S is the surface of the paraboloid z = 1-x2-y2 corresponding to 0, and n is a unit normal vector to S in the positive z-direction. (a) Apply Stokes' theorem to evaluate the integral. b) Evaluate the integral directly over the surface S. (c) Evaluate the integral directly over the new surface S which is given by the disk
(2) Let F zi + xj+yk...
F-dS where S is the cylinder x? +-2, 0 s y < 2 oriented by the unit normal 5- Let F(x,y,z)= (-6x,0,-62). Evaluate pointing out of the cylinder. 6-Let F(x, y,2)- yi- xj +zx°y?k. Evaluate (Vx F) . dS where S is the surface x2+y+32 - 1, z <0 oriented by the upward- pointing unit normal.
F-dS where S is the cylinder x? +-2, 0 s y
Q2 13 Points Begin with the paraboloid = 22 + y2, for 0 < < 4, and slice it with the plane y 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2x + y, 2x + z, 2y + x>. ZA с S 2 =...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outward pointing normal in dS terms r and θ. This cone has an open bottom. . The integrand must be fully simplified » Do not evaluate the integral
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the...
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