Vector fields in polar coordinates A vector field in polar coordinates has the form F(r, θ) = F(r, θ) ur + g(r, θ) uθ, where the unit
62. Vectors in ℝ2 may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted ur and uθ (see figure). Unlike the coordinate unite vectors in Cartesian coordinates, ur and uθ 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:
ur = cos θi + sin θj i = ur cos θ − uθ sin θ
uθ = sin θi + cos θj j = ur sin θ + uθ cos θ
66. F = r uθ
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