Homework Answers
te tt 9 Let S be the surface defined by r2 y-22= 1 and 0 15...
te tt 9 Let S be the surface defined by r2 y-22= 1 and 0 15 points by the normal direction toward the z-axis. Find the flux of the velocity field V 1and oriented (z2-ry2)i+(2.2y -yz2)j+ (y2z- 2r2)k across S Solution. To use Gauss Theorem, define C and C2 such that C1 {(a, y, z ) | a + y? < 2, z = 1} C2={(r,y,)| +y1.0} 11 TALK X W P S ww F6 & # 2 3 4...
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...
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. You are given the following multivariate PDF 3 (x, y, z) else s fxx.2(z, y, z)- I, 0 where S-(...
2. You are given the following multivariate PDF 3 (x, y, z) else s fxx.2(z, y, z)- I, 0 where S-((z, y,2)lr'ザ+8-1) (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the probabilities P(X,Y, Z)...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given ...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y,...
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0, 2 0} S = = Identify a parametrization d: U -> S of S (so UC R2 open so that S is part of a cone. etc.) such that d 1 is a conformal chart Suggestion: parametrize as a surface of revolution.
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0,...
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e ...
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z)
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z)
Let S be the surface of the box given by {(x, y, z) – 2 <<<0,...
Let S be the surface of the box given by {(x, y, z) – 2 <<<0, -1<y<2, 0<z<3} with outward orientation. Let Ę =< -æln(yz), yln(yz), –22 > be a vector field in R3. Using the Divergence Theorem, compute the flux of F across S. That is, use the Divergence Theorem to compute SS F. ds S
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0...
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0 else where S-((x, y, z) 1x2 + y2 +#51). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the...
You are given the following multivariate PDF (x, y, z) ES fxx.2(x, y, z) =- 0 else where S-((z, y...
You are given the following multivariate PDF (x, y, z) ES fxx.2(x, y, z) =- 0 else where S-((z, y, z) 1x2 + уг + z2 < 1} (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S....
te tt 9 Let S be the surface defined by r2 y-22= 1 and 0 15 points by the normal direction toward the z-axis. Find the flux of the velocity field V 1and oriented (z2-ry2)i+(2.2y -yz2)j+ (y2z- 2r2)k across S Solution. To use Gauss Theorem, define C and C2 such that C1 {(a, y, z ) | a + y? < 2, z = 1} C2={(r,y,)| +y1.0} 11 TALK X W P S ww F6 & # 2 3 4...
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...
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. You are given the following multivariate PDF 3 (x, y, z) else s fxx.2(z, y, z)- I, 0 where S-((z, y,2)lr'ザ+8-1) (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the probabilities P(X,Y, Z)...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0, 2 0} S = = Identify a parametrization d: U -> S of S (so UC R2 open so that S is part of a cone. etc.) such that d 1 is a conformal chart Suggestion: parametrize as a surface of revolution.
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0,...
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z)
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z)
Let S be the surface of the box given by {(x, y, z) – 2 <<<0, -1<y<2, 0<z<3} with outward orientation. Let Ę =< -æln(yz), yln(yz), –22 > be a vector field in R3. Using the Divergence Theorem, compute the flux of F across S. That is, use the Divergence Theorem to compute SS F. ds S
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0 else where S-((x, y, z) 1x2 + y2 +#51). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the...
You are given the following multivariate PDF (x, y, z) ES fxx.2(x, y, z) =- 0 else where S-((z, y, z) 1x2 + уг + z2 < 1} (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S....
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