Find the divergence of the following functions in cylindrical
coordinate systems:
(a) r cosθ ˆ er − r sinθ ˆ eθ
(b) 1/ r ˆ er
Find the divergence of the following functions in cylindrical coordinate systems: (a) r cosθ ˆ er...
1 Cylindrical coordinate system Given the relation of the cylindrical coordinate system r=r cos pi+r sin øj + zk (1) Lets define vectors er, eg, and ez, that indicate the direction of the vectors in the cylindrical coordinate system. Using the definition ar e = pt=r, p2 = 0, p3 (2) (a) Find a matrix for calculating er, er and e, in terms of i, j, and k. Invert the relation for expressing i, j, and k in terms of...
Q.1 a) Calculate the scale factors for the cylindrical and spherical coordinate systems. b) Calculate the Cartesian unit vectors in terms of the cylindrical and spherical unit vectors. c) Calculate the cylindrical and spherical unit vectors in terms of the Cartesian unit vectors.
6. Find the derivative matrices for the change-of-coordinate functions, then find their determinants! (a) f(r,0)= (r cos 0, r sin 0) (b) f(r,0,2) (r cos 0, r sin 0, ) (c) f(p,0,)(psin o cos 9, psin o sin 0, p cos o) 6. Find the derivative matrices for the change-of-coordinate functions, then find their determinants! (a) f(r,0)= (r cos 0, r sin 0) (b) f(r,0,2) (r cos 0, r sin 0, ) (c) f(p,0,)(psin o cos 9, psin o sin...
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...
be a coordinate function for 1.) (Coordinate functions) Let f: R a (a) (Exercise 3A) Prove that -f is a coordinate function as well. (5 (b) For which real numbers a, b e R is a f+b a coordinate function? (c) Let g:E → R be a coordinate function. Prove that there exists a line ( points) Justify your answer with a proof. (10 points) real number b E R, such that g-f+b or gf+b. (10 points)
1. Find the divergence, curl and Laplacian of the following vector fields (a) E = psin o Ô-p?Ộ - zk, where p, 0, z are cylindrical coordinates. (b) F = sin O † – rsin e ôn, where r, 0, $ are spherical coordinates.
In cylindrical coordinates (r, , z), a torus (a.k.a. the mathematical doughnut) has the equation Change the coordinate system from cylindrical coordinates (r, , z) to torodial coordinates () where Find the surface area of the torus. We were unable to transcribe this imager-a We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image r-a
Find the divergence Find the divergence of the following vector field: E = x+_y + _z where b is a constant + r
(5) a) Sketch r = 3+ 3 cosθ and b) Find the are length of the curve for 2π/3 ≤ θ ≤ π
Let g: R3 R3 be the cylindrical coordinate transformation g(r, 0, 2) (r cos e,r sin 0, 2). Which of the following is equal to det(Dg(r, 0, 2))? Ο η r cos(0) O-rsin(O)