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1. (5 pts.) TRue or FALse: (a) Let R denote a plane region, and (u,u) = (u(x,y), u(x,y)) be a dif...
1. (5 pts.) True oR FALSE: (a) Let R denote a plane region, and (u, v) - (u(x, y), v(x, y)) be a ...
1. (5 pts.) True oR FALSE: (a) Let R denote a plane region, and (u, v) - (u(x, y), v(x, y)) be a different set of coordinates for the Cartesian plane. Then for any function F(u, v) F(u, v)dudv-F(u(x, y), v(x, y))drdy (b) Let R denote a plane region, and (u,v) (u(x,y),o(x,y)) be a different set of coordinates for the Cartesian plane. Then dudv (c) Let R denote a square of sidelength 2 defined by the inequalities r S1, ly...
Ework assignments. Please see the course syllabus for This one is due at the beginning 1. (5 pts....
ework assignments. Please see the course syllabus for This one is due at the beginning 1. (5 pts.) TRUE OR FALSE: (a) Let R denote a plane region, and (u,v) (u(,y),v)) be a different set of coordinates for the Cartesian plane. Then for any function F(u,v) F(u, v)dudvF(u(r, y),v(x, y))dzdy (b) Let R denote a plane region, and (u, v) (u(r, w),v(x, y)) be a different set of coordinates for the Cartsian plane. Then do (c) Let R denote a...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
(c) Let F be the vector field on R given by F(x, y, z) = (2x...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0 otherwise ...
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0 otherwise Show that the joint density function of U = 3(X-Y) and V = Y is otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fu (11) te-lul foru R.
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0...
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1)...
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x, y, 0) = (0, y, a) orientation-preserving or
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x,...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y)...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
2. Let f(x,y) = e In(y) and let R be the region in the first quadrant...
2. Let f(x,y) = e In(y) and let R be the region in the first quadrant of the plane that lies above r = = In(y) from y=1 to y = 2. (a) Sketch the region R in the plane. (b) Evaluate SSR f(x,y) dA.
Q5. (a) Consider the region in the complex plane defined by: z = x+iy : 1,...
Q5. (a) Consider the region in the complex plane defined by: z = x+iy : 1, lul π/3. Draw this region in the z-plane and mark a few points on it of your choice (eg, A, B, C) Now, apply the conformal transformation w-e*. Plot the resulting region and mark the corresponding points (eg., A, B, C.) (b) What is the area (in arbitrary square units) of the figure in the z-plane? What is the area in the w-plane?
1. (5 pts.) True oR FALSE: (a) Let R denote a plane region, and (u, v) - (u(x, y), v(x, y)) be a different set of coordinates for the Cartesian plane. Then for any function F(u, v) F(u, v)dudv-F(u(x, y), v(x, y))drdy (b) Let R denote a plane region, and (u,v) (u(x,y),o(x,y)) be a different set of coordinates for the Cartesian plane. Then dudv (c) Let R denote a square of sidelength 2 defined by the inequalities r S1, ly...
ework assignments. Please see the course syllabus for This one is due at the beginning 1. (5 pts.) TRUE OR FALSE: (a) Let R denote a plane region, and (u,v) (u(,y),v)) be a different set of coordinates for the Cartesian plane. Then for any function F(u,v) F(u, v)dudvF(u(r, y),v(x, y))dzdy (b) Let R denote a plane region, and (u, v) (u(r, w),v(x, y)) be a different set of coordinates for the Cartsian plane. Then do (c) Let R denote a...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0 otherwise Show that the joint density function of U = 3(X-Y) and V = Y is otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fu (11) te-lul foru R.
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0...
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x, y, 0) = (0, y, a) orientation-preserving or
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x,...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
2. Let f(x,y) = e In(y) and let R be the region in the first quadrant of the plane that lies above r = = In(y) from y=1 to y = 2. (a) Sketch the region R in the plane. (b) Evaluate SSR f(x,y) dA.
Q5. (a) Consider the region in the complex plane defined by: z = x+iy : 1, lul π/3. Draw this region in the z-plane and mark a few points on it of your choice (eg, A, B, C) Now, apply the conformal transformation w-e*. Plot the resulting region and mark the corresponding points (eg., A, B, C.) (b) What is the area (in arbitrary square units) of the figure in the z-plane? What is the area in the w-plane?
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