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Evaluate f Hall of Stoke's theorem for the field H-10sinθαθ and the path enclosing the surface...
Problem 3. Vector Calculus (30 points) Set up the equations for Stoke's theorem for the vector...
Problem 3. Vector Calculus (30 points) Set up the equations for Stoke's theorem for the vector field B frcosd + psinp in the surface shown below. No need to evaluate the equations (a) Write the closed contour integral equation and simplify any vector algebra (gradient, divergence, curl, dot product, etc.). Do not evaluate the integral. (15 points) (b) Write the open surface integral equation and simplify any vector algebra (gradient, divergence, curl, dot product, etc.). Do not evaluate the integral....
4. Stoke's Theorem: Consider a vector field F = (1,1)+(1,0) + (0,0). tyle +rin the unit...
4. Stoke's Theorem: Consider a vector field F = (1,1)+(1,0) + (0,0). tyle +rin the unit square bordered by (0,0) + (0,1) ► (a) What is the curl of the vector field F? [2 points) (b) What is the path integral of the vector field around the unit square? [5 points) (c) Show your answers to the previous parts are consistent with Stoke's Theorem. HINT: consider the right-hand rule. (3 points]
a) Complete the statement of: Stoke's Theorem: Let S be an oriented surface bounded by a...
a) Complete the statement of: Stoke's Theorem: Let S be an oriented surface bounded by a piecewise smooth simple closed curve with a positive orientation (i.e. clockwise relative to N). If F(x, y, z)=(M(x,y,z), N(x, y, z), P(x, y, z)) where M, N, and P have continuous partial derivatives in an open region containing Sand C, then: b) Use Stoke's theorem to write as an iterated integral, J. (y, -2', 1)odr where is the circle of radius 1 in the...
2. Consider the vector field F = (z v)a I zy (z + a)2. Consider also...
2. Consider the vector field F = (z v)a I zy (z + a)2. Consider also a frustum of cone defined as: (see figure). Let us call V the volume of this solid. Alio, let S be the closed surface enclosing the volume: S -S1 U S2 U S3, where Si is the flat bottom (z = 1), S2 is the curved surface and Ss is the flat top (z 4). (a) calculate the flux Ф-ISF ds, using the appropriate...
Help with question 2 1. what is the electric field at the centre (r = 0)...
Help with question 2
1. what is the electric field at the centre (r = 0) of a hemisphere bounded by r = a, 0 < θ < π/2 and 0 < φ < 2m, that carries a uniform volumetric charge density P3(The charges are distributed in this hemispherical 3D space. Use spherical coordinates due to the hemispherical geometry.) Consider some charges that are lined up in a straight line. This line of charge carries a uniform linear charge density....
2. Given the vector field F-ki/r+zk22, evaluate the scalar surface integral (1) over the surface of...
2. Given the vector field F-ki/r+zk22, evaluate the scalar surface integral (1) over the surface of a closed cylinder about the z-axis specified by 2 = +3 and r = 2, as described in Fig. 1, where ki and ky are constants. Fig. 1. A cylindrical surface.
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as...
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original...
(1 point) Compute the flux of the vector field F 3z2y2 zk through the surface S...
(1 point) Compute the flux of the vector field F 3z2y2 zk through the surface S which is the cone vz2 y2 z, with 0z R, oriented downward. (a) Parameterize the cone using cylindrical coordinates (write 0 as theta). (r,)cos(theta) (r, e)rsin(theta) witho KTR and 0 (b) With this parameterization, what is dA? dA = | <0,0,(m5/2)sin^2(theta» (c) Find the flux of F through S flux
2. (8 points) Let F- try - vy2. Evaluate the line integralF dl using the f...
2. (8 points) Let F- try - vy2. Evaluate the line integralF dl using the f ollowing paths: a) (2 points) Going from 0, 0 to r 2,y 1 along the straight line y b) (2 points) Going from 0, 0 to r 2,y-1 along the parabola y c) (2 points) Going from 0,y 0 to 2,y 1 by going first along a vertical line from0,y0 to x-0, y-l then along the horizontal line x , y-i to x-2, y-1,...
1. what is the electric field at the centre (r = 0) of a hemisphere bounded...
1. what is the electric field at the centre (r = 0) of a hemisphere bounded by r = a, 0 < θ < π/2 and 0 < φ < 2m, that carries a uniform volumetric charge density P3(The charges are distributed in this hemispherical 3D space. Use spherical coordinates due to the hemispherical geometry.) Consider some charges that are lined up in a straight line. This line of charge carries a uniform linear charge density. Let's make Pl =...
Problem 3. Vector Calculus (30 points) Set up the equations for Stoke's theorem for the vector field B frcosd + psinp in the surface shown below. No need to evaluate the equations (a) Write the closed contour integral equation and simplify any vector algebra (gradient, divergence, curl, dot product, etc.). Do not evaluate the integral. (15 points) (b) Write the open surface integral equation and simplify any vector algebra (gradient, divergence, curl, dot product, etc.). Do not evaluate the integral....
4. Stoke's Theorem: Consider a vector field F = (1,1)+(1,0) + (0,0). tyle +rin the unit square bordered by (0,0) + (0,1) ► (a) What is the curl of the vector field F? [2 points) (b) What is the path integral of the vector field around the unit square? [5 points) (c) Show your answers to the previous parts are consistent with Stoke's Theorem. HINT: consider the right-hand rule. (3 points]
a) Complete the statement of: Stoke's Theorem: Let S be an oriented surface bounded by a piecewise smooth simple closed curve with a positive orientation (i.e. clockwise relative to N). If F(x, y, z)=(M(x,y,z), N(x, y, z), P(x, y, z)) where M, N, and P have continuous partial derivatives in an open region containing Sand C, then: b) Use Stoke's theorem to write as an iterated integral, J. (y, -2', 1)odr where is the circle of radius 1 in the...
2. Consider the vector field F = (z v)a I zy (z + a)2. Consider also a frustum of cone defined as: (see figure). Let us call V the volume of this solid. Alio, let S be the closed surface enclosing the volume: S -S1 U S2 U S3, where Si is the flat bottom (z = 1), S2 is the curved surface and Ss is the flat top (z 4). (a) calculate the flux Ф-ISF ds, using the appropriate...
Help with question 2
1. what is the electric field at the centre (r = 0) of a hemisphere bounded by r = a, 0 < θ < π/2 and 0 < φ < 2m, that carries a uniform volumetric charge density P3(The charges are distributed in this hemispherical 3D space. Use spherical coordinates due to the hemispherical geometry.) Consider some charges that are lined up in a straight line. This line of charge carries a uniform linear charge density....
2. Given the vector field F-ki/r+zk22, evaluate the scalar surface integral (1) over the surface of a closed cylinder about the z-axis specified by 2 = +3 and r = 2, as described in Fig. 1, where ki and ky are constants. Fig. 1. A cylindrical surface.
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original...
(1 point) Compute the flux of the vector field F 3z2y2 zk through the surface S which is the cone vz2 y2 z, with 0z R, oriented downward. (a) Parameterize the cone using cylindrical coordinates (write 0 as theta). (r,)cos(theta) (r, e)rsin(theta) witho KTR and 0 (b) With this parameterization, what is dA? dA = | <0,0,(m5/2)sin^2(theta» (c) Find the flux of F through S flux
2. (8 points) Let F- try - vy2. Evaluate the line integralF dl using the f ollowing paths: a) (2 points) Going from 0, 0 to r 2,y 1 along the straight line y b) (2 points) Going from 0, 0 to r 2,y-1 along the parabola y c) (2 points) Going from 0,y 0 to 2,y 1 by going first along a vertical line from0,y0 to x-0, y-l then along the horizontal line x , y-i to x-2, y-1,...
1. what is the electric field at the centre (r = 0) of a hemisphere bounded by r = a, 0 < θ < π/2 and 0 < φ < 2m, that carries a uniform volumetric charge density P3(The charges are distributed in this hemispherical 3D space. Use spherical coordinates due to the hemispherical geometry.) Consider some charges that are lined up in a straight line. This line of charge carries a uniform linear charge density. Let's make Pl =...
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