Unformatted text preview: ϕ in the direction of most rapid increase at the point (0 , 1 , 2)? [3 pts.] 4. Derive the identity ∇ · ( f v ) = ∇ f · v + f ∇ · v , (1) where f is a scalar ﬁeld and v is a vector ﬁeld. [5 pts.] 5. Evaluate the expression ∇ · r , (2) where r is the position vector from the origin to the point r , using (i) Cartesian coordinates and (ii) spherical coordinates (cf. (6.32)). [3+2=5 pts.] 6. (a) Find the most general function f ( r ) of only the radial coordinate r =  r  such that ∇ · ( f r ) = 0 except possibly at the origin r = 0. (b) Find the most general function g ( r,θ,ϕ ) such that ∇ · ( g r ) = 0, again except possibly at the origin r = 0. [4+1=5 pts.] 7. Problems 10.1a,b,c,d. ( Curvature of a function ) [2+1+4+3=10 pts.] 8. Problem 10.1 g ( Stability of equilibrium ) [5 pts.]...
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 Fall '10
 Jacobson
 Physics, Force, pts, Polar coordinate system, Prof. Ted Jacobson

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