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Unformatted text preview: Mathematics 20E  Spring 2009  Homework 5
June 9, 2009
Please complete the following problems and turn them in to the homework drop box in the sixth floor of AP&M labelled "Math 20E / Ryan Szypowski" by Friday, June 5 at 5:30pm. Only a subset of the problems will be graded; leave some blank at your own risk. 1. In this problem, we will show that a particular vector field is path independant from first principles without invoking any theorems about conservative vector fields. Consider the constant vector field F = i + j + k. Let C1 and C2 be any two oriented curves with the same starting point and ending point. That is, let c1 : [a1 , b1 ] 3 and c2 : [a2 , b2 ] 3 be parameterizations of C1 and C2 respectively. Then, knowing only that c1 (a1 ) = c2 (a2 ) and c1 (b1 ) = c2 (b2 ), show F ds =
C1 C2 F ds. 2. Recall that if F(x, y, z) = F1 (z, y, z)i + F2 (x, y, z)j + F3 (x, y, z)k satisfies F = 0 then F = G for some G. Prove that the following 1 construction of G works: G(x, y, z) = G1 (x, y, z)i + G2 (x, y, z)j + G3 (x, y, z)k
z y G1 (x, y, z) =
0 F2 (x, y, t) dt 
0 z F3 (x, t, 0) dt G2 (x, y, z) = 
0 F1 (x, y, t) dt G3 (x, y, z) = 0. Note however, that this construction is clearly not unique since we could add the gradient of any function to G and still achieve this result. 3. First, show that the vector field F = (x2 + 1)i + (z  2xy)j + yk satisfies F = 0. Then, use the above construction to find G such that F = G. Solution: Notice that F = 2 (x + 1) + (z  2xy) + (y) x y z = 2x  2x + 0 = 0. Also, F is a nice, smooth function everywhere. Thus, F = G for 2 some vector field G. In order to find, G, we use the above construction:
z y G1 (x, y, z) =
0 z F2 (x, y, t) dt 
0 y F3 (x, t, 0) dt t dt =
0 t  2xy dt 
0 = 1 1 2 z  2xyz  y 2 , 2 2
z G2 (x, y, z) = 
0 z F1 (x, y, t) dt x2 + 1 dt
0 =  = x2 z  z G3 (x, y, z) = 0. 4. Determine which of the following vector fields is the gradient of some scalar field, and if it is, find the scalar field: (a) F(x, y, z) = xi + yj + zk. Solution: In order to determine if F is the gradient of some vector field, we can simply compute F and see if it is 0. Here, i F = = 0. Thus, F is the gradient of some scalar field. To find it, we use the construction
x y z x j y k z x y z f (x, y, z) =
0 x F1 (t, 0, 0) dt +
0 y F2 (x, t, 0) dt +
0 z F3 (x, y, t) dt =
0 t dt +
0 t dt +
0 t dt = 1 2 (x + y 2 + z 2 ). 2 3 (b) F(x, y, z) = xyz(i + j + k). Solution: In order to determine if F is the gradient of some vector field, we can simply compute F and see if it is 0. Here, i F = x j y k z xyz xyz xyz = (xz  xy)i + (xy  yz)j + (yz  xz)k. Thus, F is not the gradient of some scalar field. (c) F(x, y, z) = (x2 + y 2 + z 2 )i + 2xyj + 2xzk. Solution: In order to determine if F is the gradient of some vector field, we can simply compute F and see if it is 0. Here, i F = x 2 j y k z x2 + y + z 2 2xy 2xz = 0. Thus, F is the gradient of some scalar field. To find it, we use the construction
x y z f (x, y, z) =
0 x F1 (t, 0, 0) dt +
0 y F2 (x, t, 0) dt +
0 z F3 (x, y, t) dt =
0 t2 dt +
0 2xt dt +
0 2xt dt = 1 3 x + xy 2 + xz 2 . 3 5. Verify Gauss' theorem by computing both F dV
V 4 and F dS
V where V = [0, 1] [0, 1] [0, 1] and F = x2 i + y 2j + z 2 k. It is probably easiest to compute the surface integral by recognizing the integral over each of the 6 faces as some particular area integral. Solution: For the first part, we simply compute
1 1 0 1 1 0 1 F dV
V =
0 2x + 2y + 2z dx dy dz x2 + 2xy + 2xz
0 1 0 1 1 x=0 = =
0 1 0 dy dz 1 + 2y + 2z dy dz y + y 2 + 2yz
0 1 1 y=0 = =
0 dz 2 + 2z dz
1 z=0 = 2z + z 2 = 3. For the scond, we split V up into 6 faces, and compute the surface integral over each. Here, I'll show the details for 2 faces: the face F1 = {(x, y, z) : x = 0, 0 y, z 1} and F2 = {(x, y, z) : x = 1, 0 y, z 1}. For F1 , recognise that the unit outward normal is n = (1, 0, 0). Thus, (x2 , y 2, z 2 ) dS =
F1 F1 (x2 , y 2, z 2 ) ndS x2 dS.
F1 = 5 However, x = 0 on F1 , and so this integral will simply be 0. For F2 , recognise that the outward normal is n = (1, 0, 0). Thus, (x2 , y 2, z 2 ) dS =
F2 F2 (x2 , y 2, z 2 ) ndS x2 dS.
F2 = On F2 , x = 1, and so this will simply compute the surface area of F2 which is 1. The other faces can be done similarly and we arrive at a total integral of 3. 6. Compute the surface integral of F = x3 i + y 3 j + z 3 k over the surface of the unit sphere, S. That is, find F dS.
S Solution: Let V be the unit ball whose boundary is S, the unit sphere. We now use Gauss' theorem to translate this into F dS =
S V F dV 3x2 + 3y 2 + 3z 2 dV.
V = This integral can be tackled easily by using spherical coordinates, 2 0 2 0 0 1 3x + 3y + 3z dV
V 2 2 2 =
0 3r 4 sin dr d d 3 sin d d 5 =
0 =
0 6 sin d 5 12 = . 5 6 7. Let W = [0, 1] [0, 1] [0, 1] and F = (x + y + z)i + (y + z 2 )j + zk. Compute F dS.
W Solution: We use Gauss' theorem and compute F dS =
W W 1 F dV
1 0 0 1 =
0 3 dV = 3. 7 ...
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This note was uploaded on 10/22/2009 for the course CHEM 140A taught by Professor Whiteshell during the Spring '04 term at UCSD.
 Spring '04
 Whiteshell

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