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T aralar toop looted at y q,z o carries a direct curvent of I0A atlong ap. Dejemine H at points (...
1. (15 points) For any parameter t, show that the R-K method (ks = /(z,-+ (1-1)h,y,, + (1-1) ). has the local truncation error O(h3) 1. (15 points) For any parameter t, show that the R-K...
1. (15 points) For any parameter t, show that the R-K method (ks = /(z,-+ (1-1)h,y,, + (1-1) ). has the local truncation error O(h3)
1. (15 points) For any parameter t, show that the R-K method (ks = /(z,-+ (1-1)h,y,, + (1-1) ). has the local truncation error O(h3)
1. A time-dependent electromagnetic field is given by, Ē(r,t = 0) = ĉEoe-(z/a)?, H (r,t=0) =...
1. A time-dependent electromagnetic field is given by, Ē(r,t = 0) = ĉEoe-(z/a)?, H (r,t=0) = 0 The field is located in vacuum, in an infinite three dimensional space. (a) Evaluate aĒ(r, t)/at and a (r, t)/at at t = 0. (b) Find the values of the fields, Ē(r, t) and H (r, t) for a general time t, satisfying ct >> a. (c) Sketch the fields found in (b) and interpret them physically.
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z)...
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z) = 1300e-x-2y-2? where T is measured in °C and x, y, and z in meters. 1. Find the rate of change of the temperature at the point P(2, -1, 2) in the direction toward the point Q(3,-3,3). Answer: Dp S(2.-1, 2) = 2. In what direction does the temperature increase fastest at P? Answer: 3. Find the maximum rate of increase at P. Answer:
6. (10 points) (a) (6 points) The gradient of the function o(x, y, z) at (1,2,3)...
6. (10 points) (a) (6 points) The gradient of the function o(x, y, z) at (1,2,3) is the vector (2, 1, 1) and g(1,2,3) = 1 (1) (2 points) Find the equation of the tangent plane of the level surface g(r, y, z) = 1 at (1,2,3) (ii) (2 points) Find the maximum rate of change of g(x, y, z) at (1, 2, 3). hax. rarte ot change: 23 14 (iii) (2 points) Find the rate of change of g...
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be ...
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be x =- (a) Find a point P-(zo, 30, zo) where F is parallel to 1. Find a point Q (which F and I are perpendicular. Q= and l are perpendicular Give an equation for the set of all points at which F and l are perpendicular. equation:
(I point) Let F=21+(z...
The quantities 2,y,z and t satisfy z= f (2,y), 1 = g(t) et y=h(t). Given g(3)...
The quantities 2,y,z and t satisfy z= f (2,y), 1 = g(t) et y=h(t). Given g(3) = -2 gl (3) = 3 h (3) = -2 hi (3) = -1, fic (-2,-2) = 3 fly (-2,-2) = 2 dz compute at t = 3 dt
If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = ...
If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = (x, z)y for all show the following: sional range, show tha x H. Hence 0 (b) T-AT, λ is a scalar.
If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = (x, z)y for all show the following: sional range, show tha x H. Hence 0 (b) T-AT,...
Solve the problem 6 Hint- Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z)...
Solve the problem 6
Hint-
Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z) and B-( (z, y) є Q : y2 z). Let also f be a real-valued integrable function on such that AfdV 4. lem 6. Let (i) If Jo/dV = 3 find fBfdV, and compute the value of JB(2f + 5)dV. Hint: use the Tesult of problem 5 (ii) If f > 0 on A and E c A such that Vol(A \ E) =...
For b.), it is from 20 to -20. Not 10 to -10 3. (40 points) Consider the time signals shown in Figure3 h(t) 10 z(t) 2 -10 Figure 3 Find y(t)-h(t)sz(t) using the graphical approach of the convolution i...
For b.), it is from 20 to -20.
Not 10 to -10
3. (40 points) Consider the time signals shown in Figure3 h(t) 10 z(t) 2 -10 Figure 3 Find y(t)-h(t)sz(t) using the graphical approach of the convolution integral (by hand). You can use MATLAB to ver
3. (40 points) Consider the time signals shown in Figure3 h(t) 10 z(t) 2 -10 Figure 3 Find y(t)-h(t)sz(t) using the graphical approach of the convolution integral (by hand). You can use MATLAB...
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. (15 points) For any parameter t, show that the R-K method (ks = /(z,-+ (1-1)h,y,, + (1-1) ). has the local truncation error O(h3)
1. (15 points) For any parameter t, show that the R-K method (ks = /(z,-+ (1-1)h,y,, + (1-1) ). has the local truncation error O(h3)
1. A time-dependent electromagnetic field is given by, Ē(r,t = 0) = ĉEoe-(z/a)?, H (r,t=0) = 0 The field is located in vacuum, in an infinite three dimensional space. (a) Evaluate aĒ(r, t)/at and a (r, t)/at at t = 0. (b) Find the values of the fields, Ē(r, t) and H (r, t) for a general time t, satisfying ct >> a. (c) Sketch the fields found in (b) and interpret them physically.
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z) = 1300e-x-2y-2? where T is measured in °C and x, y, and z in meters. 1. Find the rate of change of the temperature at the point P(2, -1, 2) in the direction toward the point Q(3,-3,3). Answer: Dp S(2.-1, 2) = 2. In what direction does the temperature increase fastest at P? Answer: 3. Find the maximum rate of increase at P. Answer:
6. (10 points) (a) (6 points) The gradient of the function o(x, y, z) at (1,2,3) is the vector (2, 1, 1) and g(1,2,3) = 1 (1) (2 points) Find the equation of the tangent plane of the level surface g(r, y, z) = 1 at (1,2,3) (ii) (2 points) Find the maximum rate of change of g(x, y, z) at (1, 2, 3). hax. rarte ot change: 23 14 (iii) (2 points) Find the rate of change of g...
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be x =- (a) Find a point P-(zo, 30, zo) where F is parallel to 1. Find a point Q (which F and I are perpendicular. Q= and l are perpendicular Give an equation for the set of all points at which F and l are perpendicular. equation:
(I point) Let F=21+(z...
The quantities 2,y,z and t satisfy z= f (2,y), 1 = g(t) et y=h(t). Given g(3) = -2 gl (3) = 3 h (3) = -2 hi (3) = -1, fic (-2,-2) = 3 fly (-2,-2) = 2 dz compute at t = 3 dt
If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = (x, z)y for all show the following: sional range, show tha x H. Hence 0 (b) T-AT, λ is a scalar.
If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = (x, z)y for all show the following: sional range, show tha x H. Hence 0 (b) T-AT,...
Solve the problem 6
Hint-
Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z) and B-( (z, y) є Q : y2 z). Let also f be a real-valued integrable function on such that AfdV 4. lem 6. Let (i) If Jo/dV = 3 find fBfdV, and compute the value of JB(2f + 5)dV. Hint: use the Tesult of problem 5 (ii) If f > 0 on A and E c A such that Vol(A \ E) =...
For b.), it is from 20 to -20.
Not 10 to -10
3. (40 points) Consider the time signals shown in Figure3 h(t) 10 z(t) 2 -10 Figure 3 Find y(t)-h(t)sz(t) using the graphical approach of the convolution integral (by hand). You can use MATLAB to ver
3. (40 points) Consider the time signals shown in Figure3 h(t) 10 z(t) 2 -10 Figure 3 Find y(t)-h(t)sz(t) using the graphical approach of the convolution integral (by hand). You can use MATLAB...
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.
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