1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z)...
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...
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.
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...
Coordinate system in rotation. Consider the Minkowski space in spherical coordinates (t, r, θ, φ) and perform a coordinate transformation to a rotational system given by t '= t, r' = r, θ '= θ, φ' = φ + ωt. (a) Find the metric in the new coordinates and all the Christoffel symbols. (b) Take θ' = θ = π/2. Write the equations of the geodesic and compare with the equations d²xi'/ dt'² = f^i, find the value of the...
5. This problem uses cylindrical coordinates. (a) Express x, y and z in terms of unit vectors in cylindrical coordinates s, ф and г. (b) Find the divergence of the function u = s(2 + sín2ф)s + s sin φ cos φ φ + 322, [3] 13)
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...
Problem 4 The parabolic cylindrical coordinates , , u) are related to the Cartesian coordinat es (x,y, z) by the transformat ion a) The line-element in Cartesian coordinates is given by d82-dr2+dy2+d22-De- termine the lne-elemen expressed in terms of the parabolic cylindrical coordinates b) Given F = 211,2) of the equation V22) F e where F depends only nu. Find the explicit form F-x F kF c) Solve the equation fro b) to find F Useful formulas: Given any ort...
question 12 , please sketch it by your hand , do not use computer graph θ varies from 0 to 2 π. φ varies from 0 to π/4 while 0 is constant. find 9-10 Write the equation in spherical coordinates. 9. (a) :2-x2 + y2 10. (a) a-2r+y- (b) x2 +z2 = 9 (b) x + 2y+ 3:-1 11-14 Sketch the solid described by the given inequalities. 15. A solid lies above the cone:- + y and below the sphere...
2014/B5 (a) Draw skecthes to illustrate R, 0 and z coordinate curves for the case of cylindrical polar coordinates (b) Show that the gradient of a scalar field, p, can be expressed in terms of curvilinear coordinates u1, u2 and us, of an orthogonal coordinate system as where h, Idr/dul. Hence obtain a formula for Vip in cylindrical polar coordinates. (c) Evaluate dp/ds, the rate of change of φ with distance, for the field φ-R, cost) at the point R...
1) If P and θ that correspond to x andy)? If P = (3.4, 29, 13) in cartesian coordianates, what are the values of r, θ, and φ of this point in spherical polar coordinates? (r, y), x = 3.3.) 4.4, what is P in polar coordinates (i.e., what are the values of r