Chapter 17.1, Problem 66E

Vector fields in polar coordinates A vector field in polar coordinates has the form F(r, θ) = F(r, θ) u_r + g(r, θ) u_θ, where the unit vectors are defined in Exercise 62. Sketch the following vector fields and express them in Cartesian coordinates.

62. Vectors in ℝ² may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted u_r and u_θ (see figure). Unlike the coordinate unite vectors in Cartesian coordinates, u_r and u_θ change their direction depending on the point (r, θ). Use the figure to show that for r > 0, the following relationships among the unit vectors in Cartesian and polar coordinates hold:

u_r = cos θi + sin θj i = u_r cos θ − u_θ sin θ

u_θ = sin θi + cos θj j = u_r sin θ + u_θ cos θ

66. F = r u_θ