2.5.1 Resolve the spherical polar unit vectors into their Cartesian compo- nents ANS. f = sin...
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...
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.
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...
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...
2.6 Addition and Subtraction of Cartesian Vectors Solution Checking: Fi 100N F-F+A 354+(-354(8600N' tb) Unit vector acting in the direction of F (35.4/100)i (35.4/100)j -0.354i -0.354j + 0.866k (86.6/100)k 2.6 Addition and Subtraction of Cartesian Vectors Solution a1 cos (0.354) 69.30 B1 cos-0.354) 111° Y1 cos(0.866) 30.0° Using the same method, F,-(10G+ 184-2 12k)kN ь.ms
(a) In Cartesian coordinates, the unit vectors r and φ are related to the unit vectors x and y by Using these expressions, and r-rr, derive Eqs. (1.48) for Fr and Fo: F, m(ř_rơ), = 1.48: (Unlike r and ф. æ and y do not change with time.) (b) Consider a particle that feels an angular force only, of the form Fo mro. (There is nothing physical about this force; it simply makes the F-ma equations solv- able.) Show that...
3. Under the influence of a vector field a particle spirals on the surface of a unit sphere toward the (t)-t and ф(t)- uppermost pole. With its spherical angular positions parametrically defined by 24t, the particle's path can be defined t€[3m/2.2n. r(t)-sin(θ(t)) cos(d(t)) ị t sin(θ(t)) sin(φ(t))J+cos(θ(t)) k, Compute the work done by the constant vector field F(,y,z) 1 k in moving the particle along this path We were unable to transcribe this image 3. Under the influence of a...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T (2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
Consider the function r 2 cos(6) + sin(26) θ (a) By looking at the Cartesian graph, where is r 0? (For 0 21. Enter your answer using interval notation.) (b) Explain why quadrants Il and Ill of the polar graph are empty (c) How many values of θ for 0 θ satisfy r= 1? (d) The polar graph intersects the unit circle 4 times. Explain the discrepancy with you answer to part (c). Consider the function r 2 cos(6) +...