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(3) Let a > 0. In spherical coordinates, a surface is defined by r = 2a cos φ for 0 Find the volu...
Find the gradient ∇φ of the following: a) φ = (r2/a2)e-r/a (using spherical coordinates) b) φ...
Find the gradient ∇φ of the following: a) φ = (r2/a2)e-r/a (using spherical coordinates) b) φ = 2√(x2+y2+z2) (using both cartesian and spherical coordinates, after converting)
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r ...
q2 please
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r and y. (2) Let f,g : R → R be functions of one variable such that f" and g" are continuous. Show that (f"(x)-g"(y)) dydx = f(0) + g(0)-f(2)-9(2) + 2f'(2) + 2g'(0). o Jo (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2acos φ for 0 φ 1. Find the...
nc = 13 1. Find the charge in the volume defined by 1<r<2m, in the spherical...
nc = 13
1. Find the charge in the volume defined by 1<r<2m, in the spherical coordinates if pv = (No cos?0)/r* (uC/mº). 2. Given that D = 7r2 a, + Nc sin 0 ag in spherical coordinates, find the charge density. 3. Find the work done in moving a point charge Q = - 20 uC from (4,2,0)m to the origin in the field E = (x/2 + 2y) ax + Nc xay (V/m). 1
PLE 2 The point (0, 5 3 , −5) is given in rectangular coordinates. Find spherical...
PLE 2 The point (0, 5 3 , −5) is given in rectangular coordinates. Find spherical coordinates for this point. SOLUTION From the distance formula we have ρ = x2 + y2 + z2 = 0 + 75 + 25 = 10 Correct: Your answer is correct. and so these equations give the following. cos(φ) = z ρ = -1/2 Correct: Your answer is correct. φ = $$ Incorrect: Your answer is incorrect. cos(θ) = x ρ sin(φ) = θ...
3. In spherical coordinates the unit vectors r, and ф are given by (a) Compute the...
3. In spherical coordinates the unit vectors r, and ф are given by (a) Compute the cross products #x f, #x θ, PX φ, θ 0, θ >< φ, and φ >< φ. (b) Express x, y and z in terms of, О and ф. (c) Check the divergence theorern for the function u = r , using for volume the sphere of radius 13] R, centered at the origin, i.e. show that dä -JyV-üö)dr.
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
Epsilon1=2 Epsilon0 6. In the spherical region #1 (0 r a, 0 θ π ,0 φ...
Epsilon1=2 Epsilon0
6. In the spherical region #1 (0 r a, 0 θ π ,0 φ 2π ), the electric field just below the spherical surface r-din region # i (where espe ) is given by E,,10+ 20 + ф40 . Gue On the spherical surface at r -a there exists a uniform surface charge density Pso 8 Find 52 in region #2 just above r a. Hint: Let E,EE+0E2,
The magnetic field intensity in all of space is given in terms of spherical coordinates: (1...
The magnetic field intensity in all of space is given in terms
of spherical coordinates:
(1 point) The magnetic field intensity in all of space is given in terms of spherical coordinates: A/m. sin θ Use this knowledge in both parts below. (a) Find the current density (in spherical coordinates) at the point P, whose Cartesian coordinates are (z,ys) = (85,-15,-2). ANSWER: At P, J a+ ag+ ap A/m2 (b) Find the net current, I,flowing through the conical surface S...
Let us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR a...
We were unable to transcribe this imageLet us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR and S(R)-S.),respectively (a) Find the relation between V(0) and S 1) (b) Calculate the Gaussian integral 3. (c) Calculate the same integral in spherical coordinates in terms of the gamma function re)-e'd (d) Obtain the closed forms of S,,(1) and V(1) (e) Calculate r5) and S.,0), p.(1) for n-1, 2, 3. (40 points)
Let us denote...
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z)...
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z) and (p',p',z'), and Cartesian coordinates (x, y, z) and (x',y',z'), respectively. Write r - r in all three coordinate systems. Hint: Use Equation 1.2) with a r r and r and r' written in terms of appropriate unit vectors.
q2 please
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r and y. (2) Let f,g : R → R be functions of one variable such that f" and g" are continuous. Show that (f"(x)-g"(y)) dydx = f(0) + g(0)-f(2)-9(2) + 2f'(2) + 2g'(0). o Jo (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2acos φ for 0 φ 1. Find the...
nc = 13
1. Find the charge in the volume defined by 1<r<2m, in the spherical coordinates if pv = (No cos?0)/r* (uC/mº). 2. Given that D = 7r2 a, + Nc sin 0 ag in spherical coordinates, find the charge density. 3. Find the work done in moving a point charge Q = - 20 uC from (4,2,0)m to the origin in the field E = (x/2 + 2y) ax + Nc xay (V/m). 1
3. In spherical coordinates the unit vectors r, and ф are given by (a) Compute the cross products #x f, #x θ, PX φ, θ 0, θ >< φ, and φ >< φ. (b) Express x, y and z in terms of, О and ф. (c) Check the divergence theorern for the function u = r , using for volume the sphere of radius 13] R, centered at the origin, i.e. show that dä -JyV-üö)dr.
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
Epsilon1=2 Epsilon0
6. In the spherical region #1 (0 r a, 0 θ π ,0 φ 2π ), the electric field just below the spherical surface r-din region # i (where espe ) is given by E,,10+ 20 + ф40 . Gue On the spherical surface at r -a there exists a uniform surface charge density Pso 8 Find 52 in region #2 just above r a. Hint: Let E,EE+0E2,
The magnetic field intensity in all of space is given in terms
of spherical coordinates:
(1 point) The magnetic field intensity in all of space is given in terms of spherical coordinates: A/m. sin θ Use this knowledge in both parts below. (a) Find the current density (in spherical coordinates) at the point P, whose Cartesian coordinates are (z,ys) = (85,-15,-2). ANSWER: At P, J a+ ag+ ap A/m2 (b) Find the net current, I,flowing through the conical surface S...
We were unable to transcribe this imageLet us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR and S(R)-S.),respectively (a) Find the relation between V(0) and S 1) (b) Calculate the Gaussian integral 3. (c) Calculate the same integral in spherical coordinates in terms of the gamma function re)-e'd (d) Obtain the closed forms of S,,(1) and V(1) (e) Calculate r5) and S.,0), p.(1) for n-1, 2, 3. (40 points)
Let us denote...
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z) and (p',p',z'), and Cartesian coordinates (x, y, z) and (x',y',z'), respectively. Write r - r in all three coordinate systems. Hint: Use Equation 1.2) with a r r and r and r' written in terms of appropriate unit vectors.
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