Q 4. Confirm that ∇ (1/ r) = − r /r 3 where r = ||r e || and r e = xˆı + y ˆj + z kˆ = ρ eˆρ + z ...
Q 4. Confirm that ∇ (1/ r) = − r /r 3 where r = ||r e || and r e = xˆı + y ˆj + z kˆ = ρ eˆρ + z eˆz = r eˆr. Do it in (i) cartesian coordinates with ∇ ≡ ∂ ∂x ˆı + ∂ ∂y ˆ + ∂ ∂z kˆ. (ii) cylindrical coordinates with ∇ ≡ ∂ ∂ρ eˆρ + 1 ρ ∂ ∂φ eˆφ + ∂ ∂z eˆz. (iii) spherical coordinates with ∇ ≡ ∂ ∂r eˆr + 1 r ∂ ∂θ eˆθ + 1 r sin(θ) ∂ ∂φ eˆφ.
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Q 4. Confirm that ∇ (1/ r) = − r /r 3 where r = ||r e || and r e = xˆı + y ˆj + z kˆ = ρ eˆρ + z ...
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.
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) r...
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) rectangular coordinates (x,y, z); (ii) cylindrical coordinates (r, 0, 2); (iii) spherical coordinates (p, φ,0). a. Inside the sphere 2 +3+224 and above the conezV b. Inside the sphere x2 + y2 + 22-12 and above the paraboloid z 2 2 + y2. c. Inside the sphere 2,2 + y2 + z2-2 and above the surface z-(z2 + y2)1/4 d. Inside the sphere...
Log(2 - 2) (x2 y Question 2. Consider the function f(x, y, (a) What is the maximal domain of f? (...
log(2 - 2) (x2 y Question 2. Consider the function f(x, y, (a) What is the maximal domain of f? (Write your answer in set notation.) (b) Find ▽f. (c) Find the tangent hyperplnes Te2)(r, y,z) and Tao2-)f(x, y, z). Find the intersection of these two hyperplanes, and very briefly describe the intersection in words (0,1, 1) and set notation. Confirm that the point (2, 2, 1) is on this level surface, and that Vf(2, 2, 1) is (d) On...
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...
1. Express the point given in Cartesian coordinates in cylindrical coordinates (r,θ,z). (9(√3/2), 9(1/2), 1)= 2....
1. Express the point given in Cartesian coordinates in
cylindrical coordinates (r,θ,z). (9(√3/2), 9(1/2), 1)=
2. Express the point given in Cartesian coordinates in spherical
coordinates (ρ,θ,ϕ). (7/3√3,21/4,7/2) =
I know we are only supposed to post 1 per question however for
this one I have 1 part correct, I just need some help with the
rest. Please if you have the time help with question 2. Thank you
for your time and knowledge.
(1 point) Express the point given...
only do problem 3c, the second picture is the answer to problem 2, the answer I...
only do problem 3c, the second picture is the answer
to problem 2, the answer I got for 3b is -1/(r^2)
The tinction V(x, y,z) Problem 3 (20 pts). Considering the function V of problem 2, (a) Show that V can be written in spherical coordinates as V(r, θ, φ-1. (10 pts) r + θ + φ (b) The gradient of a function in spherical coordinates is VV Calculate the gradient of V in spherical coordinates. (5 pts) (e) Show...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addi...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
FIG. 2. Setup of Exercise 3 Exercise 3 The electrostatic potential of an electic dipole moment...
FIG. 2. Setup of Exercise 3 Exercise 3 The electrostatic potential of an electic dipole moment d located at the origin takes the following form d-T Tr where r is the vector joining the origin to the point X (7 is called the "position vector" in the textbook). See Fig. 2 (i) Chosing the z axis to be aligned with the electric dipole moment, express φ in terms of cartesian, cylindrical, (ii) The electric field is obtained from E-- Compute...
3. (i) Find the kinetic energy of a particle of mass m with position given by...
3. (i) Find the kinetic energy of a particle of mass m with position given by the coordinates (s, u, v), related to the ordinary Cartesian coordinates by y z = 2s + 3 + u = 2u + v = 0+03 (ii) Find the kinetic energy of a particle of mass m whose position is given in cylindrical coordinates = = r cos r sine y (iii) Find the kinetic energy of a particle of mass m with position...
(3 points) (a) The Cartesian coordinates of a point are (-1,-V3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0) of the point, wh...
(3 points) (a) The Cartesian coordinates of a point are (-1,-V3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0) of the point, where r < 0 and 0 < θ < 2π. Y= (b) The Cartesian coordinates of a point are -2,3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0)...
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.
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) rectangular coordinates (x,y, z); (ii) cylindrical coordinates (r, 0, 2); (iii) spherical coordinates (p, φ,0). a. Inside the sphere 2 +3+224 and above the conezV b. Inside the sphere x2 + y2 + 22-12 and above the paraboloid z 2 2 + y2. c. Inside the sphere 2,2 + y2 + z2-2 and above the surface z-(z2 + y2)1/4 d. Inside the sphere...
log(2 - 2) (x2 y Question 2. Consider the function f(x, y, (a) What is the maximal domain of f? (Write your answer in set notation.) (b) Find ▽f. (c) Find the tangent hyperplnes Te2)(r, y,z) and Tao2-)f(x, y, z). Find the intersection of these two hyperplanes, and very briefly describe the intersection in words (0,1, 1) and set notation. Confirm that the point (2, 2, 1) is on this level surface, and that Vf(2, 2, 1) is (d) On...
1. Express the point given in Cartesian coordinates in
cylindrical coordinates (r,θ,z). (9(√3/2), 9(1/2), 1)=
2. Express the point given in Cartesian coordinates in spherical
coordinates (ρ,θ,ϕ). (7/3√3,21/4,7/2) =
I know we are only supposed to post 1 per question however for
this one I have 1 part correct, I just need some help with the
rest. Please if you have the time help with question 2. Thank you
for your time and knowledge.
(1 point) Express the point given...
only do problem 3c, the second picture is the answer
to problem 2, the answer I got for 3b is -1/(r^2)
The tinction V(x, y,z) Problem 3 (20 pts). Considering the function V of problem 2, (a) Show that V can be written in spherical coordinates as V(r, θ, φ-1. (10 pts) r + θ + φ (b) The gradient of a function in spherical coordinates is VV Calculate the gradient of V in spherical coordinates. (5 pts) (e) Show...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
FIG. 2. Setup of Exercise 3 Exercise 3 The electrostatic potential of an electic dipole moment d located at the origin takes the following form d-T Tr where r is the vector joining the origin to the point X (7 is called the "position vector" in the textbook). See Fig. 2 (i) Chosing the z axis to be aligned with the electric dipole moment, express φ in terms of cartesian, cylindrical, (ii) The electric field is obtained from E-- Compute...
3. (i) Find the kinetic energy of a particle of mass m with position given by the coordinates (s, u, v), related to the ordinary Cartesian coordinates by y z = 2s + 3 + u = 2u + v = 0+03 (ii) Find the kinetic energy of a particle of mass m whose position is given in cylindrical coordinates = = r cos r sine y (iii) Find the kinetic energy of a particle of mass m with position...
(3 points) (a) The Cartesian coordinates of a point are (-1,-V3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0) of the point, where r < 0 and 0 < θ < 2π. Y= (b) The Cartesian coordinates of a point are -2,3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0)...
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