Question

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 = φ × r. Once again, explain why this formula works. Your final result should look like this C. Go ahead and invert the above relations and write z, y, and z in terms of ŕ.0, and φ D. In spherical coordinates, the unit vectors r, θ, and φ vary from point to point. For future reference, we will want to know their derivatives. Compute the derivatives of the vectors r, θ, and φ with respect to the spherical coordinates r, θ, and φ E. Consider a change dr of the vector r. Write dr= d(rr) rar + rd(r) in spherical coordinates. G. We will compute the operator V- +joy + 20 in spherical coordinates. One way to do this is to brute force it. Here's a prettier way that requires less algebra (believe it or not, this homework is less algebra than the brute force method!). First. justify the following statement: df = ŕ . (Vf). Now, for any function f, the chain rule implies Write f-(Vf),f + (Vf),0 + (VfJa, and equate our two expressions for df, to arrive at an expression for the r, θ, and φ components of . Your final result should be rsin θ

Answer 1

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