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Unformatted text preview: HW#5 —Phys374—Spring 2008 Prof. Ted Jacobson Due before class, Friday, March 7, 2008 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374c/ [email protected] 1. Derive the identity ∇ × ( f v ) = ∇ f × v + f ∇ × v (1) where f is a scalar field and v is a vector field. [5 pts.] 2. Evaluate the expression ∇ × ( f ( r ) r ) , (2) where r is the position vector from the origin to the point r , and r = | r | , using (i) Cartesian coordinates and (ii) spherical coordinates (cf. (7.16)). In the first step, use the result of the previous problem to simplify this one. [3+2=5 pts.] 3. (i) Show that F = yz ˆ x + zx ˆ y + xy ˆ z has zero curl and divergence. Thus it can be written both as the gradient of a scalar and as the curl of a vector. (ii) Find all the scalar potentials for F (i.e. all functions Φ such that F = ∇ Φ). (iii) Find all the vector potentials for F (i.e. all vector fields H such that F = ∇ × H ). [5+5+5=15 pts.] 4. Show that if v has vanishing divergence, then v is equal to the curl of w defined by the components...
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- Fall '10
- Physics, pts, Vector field, divergenceless vector ﬁeld, Quantized superﬂuid circulation, inﬁnitesimal loop