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# rev4 - a about the origin is 4 π Use Gauss’ theorem to...

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Physics (PHZ) 3113 Florida Atlantic University Mathematical Physics Fall, 2009 Review Problems IV Recommended Reading: Chow, pp. 15–27. 1. (Chow 1.12) a. Find a unit vector normal to the surface x 2 + y 2 - z = 1 at the point (1 , 1 , 1). b. Find the directional derivative of φ ( x, y, z ) = x 2 yz + 4 xz 3 at the points (1 , - 2 , - 1) in the direction ˆ x - 2 ˆ y + 2 ˆ z . 2. (Chow 1.14) Consider the ellipse r 1 + r 2 = a , with a a constant, that is shown in Figure 1.23 of the text. Prove that the vectors r 1 and r 2 from either focus to a given point on the ellipse make equal angles with the tangent to the ellipse at that point. 3. (based on Chow 1.17) a. Show that the divergence of an inverse-square force field in three dimensions is zero, · F := · r r 3 = 0 , except possibly at the origin. b. Show that the flux of F through a sphere of radius
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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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