Homework Answers
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z)...
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of 2 (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At...
(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...
Let fand g be the functions whose graphs are shown below. 3 2 - -4 2...
Let fand g be the functions whose graphs are shown below. 3 2 - -4 2 2 3 4 5 N -2 g(x) 3 4 (a) Let u(x)=f(x)g(x). Find u'(-3). (b) Let v(x) = f(x)). Find v'(4).
(1 point) Compute the flux of the vector field F(x, y, z) = 3 + 2+...
(1 point) Compute the flux of the vector field F(x, y, z) = 3 + 2+ 2k through the rectangular region with corners at (1,1,0), (0,1,0), (0,0,2), and (1,0, 2) oriented in the positive Z-direction, as shown in the figure. 2.0 1.5 Flux = 0.0 12.0 11.5 2 1.0 0.5 0.0 2.94. god. og 9.500.00 [Enable Java to make this image interactive] (Drag to rotate) (1 point) Compute the flux of the vector field F(t, y, z) = 31 +23...
10.00 Ex vs. Time Ey = 8 Sin(t + 1.5*P1) 5.00 Ex (N/C) Z 0.00 -5.00...
10.00 Ex vs. Time Ey = 8 Sin(t + 1.5*P1) 5.00 Ex (N/C) Z 0.00 -5.00 Ex = 8 Sin(t) - 10.00 - 1.00 0.00 1.00 Time (s) 10.00 Ey vs. Time 5.00 Ey (N/C) 0.00 -5.00 - 10.00 - 1.00 Vector Sum of Ex and Ey 1.00 0.00 Time (5) Set values Ex = 8 N/CE, = 8 N/C phase difference = 1.5*n radians The animation shows the result of adding two perpendicular electric fields together. Each field is...
1 Question 3 (4 Marks) show key steps Consider the vector space M2x2(C). i Let Z...
1 Question 3 (4 Marks) show key steps Consider the vector space M2x2(C). i Let Z Span 2 + 3i 2 - 31 2i -2i Is Z? -1+i 10+ 1li s(-
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Please answer without using previously posted answers. Thanks Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a g...
Please answer without using previously posted answers.
Thanks
Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a gradient field, and φ s called a potential function. Ideal Fluid Flow Let F represent the two-dimensional velocity field of an inviscid fluid that is incompressible, ie. . F-0 (or divergence-free). F can be represented by (1), where ф is called the velocity potential-show that o is harmonic....
Consider the following vector field. F = (xi + yj + zk )/((x^2 + y^2 + z^2)^3/2) (a) Find the divergence of F. (b) Let S...
Consider the following vector field. F = (xi + yj + zk )/((x^2 + y^2 + z^2)^3/2) (a) Find the divergence of F. (b) Let S be any sphere not containing the origin. Find the outward flux of F across S. (c) Let Sa be the sphere of radius a centered at the origin. Find the outward flux of F across Sa.
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of 2 (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At...
(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...
Let fand g be the functions whose graphs are shown below. 3 2 - -4 2 2 3 4 5 N -2 g(x) 3 4 (a) Let u(x)=f(x)g(x). Find u'(-3). (b) Let v(x) = f(x)). Find v'(4).
(1 point) Compute the flux of the vector field F(x, y, z) = 3 + 2+ 2k through the rectangular region with corners at (1,1,0), (0,1,0), (0,0,2), and (1,0, 2) oriented in the positive Z-direction, as shown in the figure. 2.0 1.5 Flux = 0.0 12.0 11.5 2 1.0 0.5 0.0 2.94. god. og 9.500.00 [Enable Java to make this image interactive] (Drag to rotate) (1 point) Compute the flux of the vector field F(t, y, z) = 31 +23...
10.00 Ex vs. Time Ey = 8 Sin(t + 1.5*P1) 5.00 Ex (N/C) Z 0.00 -5.00 Ex = 8 Sin(t) - 10.00 - 1.00 0.00 1.00 Time (s) 10.00 Ey vs. Time 5.00 Ey (N/C) 0.00 -5.00 - 10.00 - 1.00 Vector Sum of Ex and Ey 1.00 0.00 Time (5) Set values Ex = 8 N/CE, = 8 N/C phase difference = 1.5*n radians The animation shows the result of adding two perpendicular electric fields together. Each field is...
1 Question 3 (4 Marks) show key steps Consider the vector space M2x2(C). i Let Z Span 2 + 3i 2 - 31 2i -2i Is Z? -1+i 10+ 1li s(-
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Please answer without using previously posted answers.
Thanks
Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a gradient field, and φ s called a potential function. Ideal Fluid Flow Let F represent the two-dimensional velocity field of an inviscid fluid that is incompressible, ie. . F-0 (or divergence-free). F can be represented by (1), where ф is called the velocity potential-show that o is harmonic....
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