Unformatted text preview: a about the origin is 4 π . Use Gauss’ theorem to show that the divergence of F cannot vanish at the origin. c. If f is a diﬀerentiable function and A is a diﬀerentiable vector ﬁeld, then show that ∇ · ( f A ) = A · ∇ f + f ∇ · A . d. Calculate the divergence of a radial vector ﬁeld (i.e., everywhere parallel to r ) in three dimensions. Under what conditions does the divergence vanish at the origin? 4. (Chow 1.20) a. Find constants a , b and c such the the vector ﬁeld A := ( x + 2 y + az ) ˆ x + ( bx-3 y-z ) ˆ y + (4 x + cy + 2 z ) ˆ z is irrotational (i.e., curl-free). b. Show that the resulting vector ﬁeld can be expressed as the gradient of a scalar ﬁeld....
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- Fall '10
- Euclidean space, Vector field, Gradient, Chow