5. This problem uses cylindrical coordinates. (a) Express x, y and z in terms of unit...
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.
106. Cylindrical Coordinates. Define curvilinear coordinates (p, ø, z) by y = p sin d, p cos integer, find expressions for the following quantities in where p 0,0 < 0 < 2t. If n is an terms of p, ф, z and p, ф, 2. (а) Vф; (b) Vр"3; (c) V2(p2 cos ); (d) V :(pp + pфф + z2); (F) V. (p*-1 sin(nф)р + pr-1 cos(nф)ф). (e) V x 106. Cylindrical Coordinates. Define curvilinear coordinates (p, ø, z) by...
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...
MARK WHICH STATEMENTS BELOW ARE TRUE, USING THE FOLLOWING, Consider Vf(x, y, z) in terms of a new coordinate system, x= x(u, v, w), y=y(u, v, w), z=z(u, v, w). Let r(s) = x(s) i+y(s) + z(s) k be the position vector defining some continuous path as a function of the arc length. Similarly for the other partial derivatives in v and w. For spherical coordinates the following must also be true for any points, x = Rsin o cose,...
Express the unit vectors, , φ in terms of 8,9, 2 (that is, derive Eq. 1.64). Check your answers several ways ( f . f = 1,0 . φ 0, f × ยิ p, you need to check all of these explicitly, the solution manual does not give all of these). Also work out the inverse formulas, giving £,9,2 in terms of f, θ and θ, φ). sn θ cos φ x + sin θ sin φ y + cos...
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.
1) Find the expressions for the unit vectors in cylindrical coordinate system, p, φ,2. in terms of x, ý, 2. Find the time-derivative of each. Hints: Unit vector p is defined in (x, y) plane. Remember that α -φ is perpendicular to a The easiest way to find φ is to express ρ through φ and add 90 degrees
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...
2. S is the surface y 2 = 4(x 2 + z 2 ), y ∈ [0, 2] obtained by rotating the function y = 2x about the y-axis for y ∈ [0, 2]. Find a suitable parametric representation of the surface S using the cylindrical polar coordinates. Answer is: 2. r(u, v) = u cos(v)i + 4uj + u sin(v)k , 0 ≤ v < 2π, 0 ≤ u ≤ 1/2. I am unsure how to work it out...
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...