374bhw4 - in the direction of most rapid increase at the...

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HW#4 —Phys374—Spring 2007 Prof. Ted Jacobson Due before class, Friday, Feb. 23, 2007 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ jacobson@physics.umd.edu 1. Problems 5.2a,b. ( Pressure gradient ) (For 5.2a, photocopy the map.) For 5.2b, you’ll have to work out or look up the distance across Ireland in the relevant direction.) [5+5=10 pts.] 2. Problems 5.2e,f,g. ( Pressure force ) Snieder basically does these for you, so all that’s left to do is just put in a few precise and concise words explaining why each equation is true. (Note ˆ x just a component of the force vector. [4+3+3=10 pts.] 3. Consider the function ϕ ( x,y,z ) = z - 3 xy . (a) Find ϕ . [3 pts.] (b) What is the rate of change of ϕ at the point (0 , 1 , 2) in the direction of the vector ˆ i + 2 ˆ j + 3 ˆ k ? ( Be careful , this is not a unit vector.) [4 pts.] (c) What is the rate of change of
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Unformatted text preview: 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 eld and v is a vector eld. [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,, ) 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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