
Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Question
Chapter 6.3, Problem 20P
a)
To determine
To find: the torque of
b)
To determine
To find: the torque of
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You manage a chemical company with 2 warehouses. The following quantities of
Important Chemical A have arrived from an international supplier at 3 different
ports:
Chemical Available (L)
Port 1
Port 2
Port 3
400
110
100
The following amounts of Important Chemical A are required at your warehouses:
Warehouse 1
Warehouse 2
Chemical Required (L)
380
230
The cost in £ to ship 1L of chemical from each port to each warehouse is as follows:
Warehouse 1 Warehouse 2
Port 1
£10
£45
Port 2
£20
£28
Port 3
£13
£11
(a) You want to know how to send these shipments as cheaply as possible. For-
mulate this as a linear program (you do not need to formulate it in standard
inequality form) indicating what each variable represents.
a) Suppose that we are carrying out the 1-phase simplex algorithm on a linear program in
standard inequality form (with 3 variables and 4 constraints) and suppose that we have
reached a point where we have obtained the following tableau. Apply one more pivot
operation, indicating the highlighted row and column and the row operations you carry
out. What can you conclude from your updated tableau?
x1 12 23
81
82
83
S4
$1
-20
1 1
0
0
0
3
82
3 0
-2
0
1
2
0
6
12
1
1
-3
0
0
1
0
2
84
-3 0
2
0
0
-1 1 4
2
-2
0 11
0
0
-4
0
-8
b) Solve the following linear program using the 2-phase simplex algorithm. You should give
the initial tableau and each further tableau produced during the execution of the
algorithm. If the program has an optimal solution, give this solution and state its
objective value. If it does not have an optimal solution, say why.
maximize 21 - - 2x2 + x3 - 4x4
subject to 2x1+x22x3x4≥ 1,
5x1+x2-x3-4 -1,
2x1+x2-x3-342,
1, 2, 3, 4 ≥0.
Suppose we have a linear program in standard equation form
maximize c'x
subject to Ax=b,
x≥ 0.
and suppose u, v, and w are all optimal solutions to this linear program.
(a) Prove that zu+v+w is an optimal solution.
(b) If you try to adapt your proof from part (a) to prove that that u+v+w
is an optimal solution, say exactly which part(s) of the proof go wrong.
(c) If you try to adapt your proof from part (a) to prove that u+v-w is an
optimal solution, say exactly which part(s) of the proof go wrong.
Chapter 6 Solutions
Mathematical Methods in the Physical Sciences
Ch. 6.3 - If A=2ijk,B=2i3j+k,C=j+k, find (AB)C,A(BC),(AB)C,...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - A force F=2i3j+k acts at the point (1,5,2). Find...Ch. 6.3 - Prob. 8PCh. 6.3 - Prob. 9PCh. 6.3 - In Figure 3.5, let r be another vector from O to...
Ch. 6.3 - Write out the twelve triple scalar products...Ch. 6.3 - (a) Simplify ( AB)2[(AB)B]A by using ( 3.9). (b)...Ch. 6.3 - Prove that the triple scalar product of (AB),(BC),...Ch. 6.3 - Prove the Jacobi identity: A(BC)+B(CA)+C(AB)=0....Ch. 6.3 - Prob. 15PCh. 6.3 - In the discussion of Figure 3.8, we found for the...Ch. 6.3 - Expand the triple product for a=(r) given in the...Ch. 6.3 - Two moving charged particles exert forces on each...Ch. 6.3 - The force F=i+3j+2k acts at the point (1,1,1). (a)...Ch. 6.3 - Prob. 20PCh. 6.4 - Verify equations (4.5) by writing out the...Ch. 6.4 - Let the position vector (with its tail at the...Ch. 6.4 - As in Problem 2, if the position vector of a...Ch. 6.4 - Prob. 4PCh. 6.4 - The position of a particle at time t is given by...Ch. 6.4 - The force acting on a moving charged particle in a...Ch. 6.4 - Sketch a figure and verify equation ( 4.12).Ch. 6.4 - In polar coordinates, the position vector of a...Ch. 6.4 - The angular momentum of a particle m is defined by...Ch. 6.4 - If V(t) is a vector function oft, find the...Ch. 6.6 - Find the gradient of w=x2y3z at (1,2,1).Ch. 6.6 - Starting from the point (1,1), in what direction...Ch. 6.6 - Find the derivative of xy2+yz at (1,1,2) in the...Ch. 6.6 - Find the derivative of zexcosy at (1,0,/3) in the...Ch. 6.6 - Find the gradient of =zsinyxz at the point...Ch. 6.6 - Find a vector normal to the surface x2+y2z=0 at...Ch. 6.6 - Find the direction of the line normal to the...Ch. 6.6 - (a) Find the directional derivative of =x2+sinyxz...Ch. 6.6 - (a) Given =x2y2z, find at (1,1,1). (b) Find the...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - Repeat Problem 14b for the following points and...Ch. 6.6 - Show by the Lagrange multiplier method that the...Ch. 6.6 - Find r, where r=x2+y2, using ( 6.7) and also using...Ch. 6.6 - As in Problem 17, find the following gradients in...Ch. 6.6 - As in Problem 17, find the following gradients in...Ch. 6.6 - As in Problem 17, find the following gradients in...Ch. 6.6 - Verify equation ( 6.8 ); that is, find f in...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Verify formulas (b), (c), (d), (g), (h), (i), (i),...Ch. 6.7 - For r=xi+yj+zk, evaluate (kr)Ch. 6.7 - For r=xi+yj+zk, evaluate rrCh. 6.7 - For r=xi+yj+zk, evaluate rrCh. 6.8 - Evaluate the line integral x2y2dx2xydy along each...Ch. 6.8 - Evaluate the line integral (x+2y)dx2xdy along each...Ch. 6.8 - Evaluate the line integral xydx+xdy from (0,0) to...Ch. 6.8 - Prob. 4PCh. 6.8 - Find the work done by the force F=x2yixy2j along...Ch. 6.8 - Prob. 6PCh. 6.8 - For the force field F=(y+z)i(x+z)j+(x+y)k, find...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Given F1=2xi2yzjy2k and F2=yixj (a) Are these...Ch. 6.8 - Which, if either, of the two force fields...Ch. 6.8 - For the force field F=yi+xj+zk, calculate the work...Ch. 6.8 - Show that the electric field...Ch. 6.8 - For motion near the surface of the earth, we...Ch. 6.8 - Consider a uniform distribution of total mass m...Ch. 6.9 - Write out the equations corresponding to ( 9.3 )...Ch. 6.9 - In Problems 2 to 5 use Greens theorem [formula (...Ch. 6.9 - In Problems 2to5useGree n stheorem[formula(9.7)]...Ch. 6.9 - In Problems 2 to 5 use Greens theorem [formula (...Ch. 6.9 - In Problems 2 to 5 use Greens theorem [formula (...Ch. 6.9 - For a simple closed curve C in the plane show by...Ch. 6.9 - Use Problem 6 to show that the area inside the...Ch. 6.9 - Use Problem 6 to find the area inside the curve...Ch. 6.9 - Apply Greens theorem with P=0,Q=12x2 to the...Ch. 6.9 - Evaluate each of the following integrals in the...Ch. 6.9 - Evaluate each of the following integrals in the...Ch. 6.9 - Evaluate each of the following integrals in the...Ch. 6.10 - Evaluate both sides of ( 10.17) if V=r=ix+jy+kz,...Ch. 6.10 - Given V=x2i+y2j+z2k, integrate Vnd over the whole...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - If F=xi+yj, calculate Fnd over the part of the...Ch. 6.10 - Evaluate Vnd over the curved surface of the...Ch. 6.10 - Given that B= curl A, use the divergence theorem...Ch. 6.10 - A cylindrical capacitor consists of two long...Ch. 6.10 - Draw a figure similar to Figure 10.6 but with q...Ch. 6.10 - Obtain Coulombs law from Gausss law by considering...Ch. 6.10 - Suppose the density of a fluid varies from point...Ch. 6.10 - The following equations are variously known as...Ch. 6.11 - Do case (b) of Example 1 above.Ch. 6.11 - Given the vector A=x2y2i+2xyj. (a) Find A (b)...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Vnd over the entire surface of the volume in the...Ch. 6.11 - (curlV)nd over the part of the surface z=9x29y2...Ch. 6.11 - Vnd over the entire surface of a cube in the first...Ch. 6.11 - Vdr around the circle (x2)2+(y3)2=9,z=0, where...Ch. 6.11 - (2xi2yj+5k)nd over the surface of a sphere of...Ch. 6.11 - (yixj+zk)dr around the circumference of the circle...Ch. 6.11 - cydx+zdy+xdz, where C is the curve of intersection...Ch. 6.11 - What is wrong with the following proof that there...Ch. 6.11 - Prob. 17PCh. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.11 - Find vector fields A such that V= curl A for each...Ch. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.12 - Prob. 1MPCh. 6.12 - If A and B are the diagonals of a parallelogram,...Ch. 6.12 - The force on a charge q moving with velocity...Ch. 6.12 - Prob. 4MPCh. 6.12 - Use Greens theorem (Section 9) to do Problem 8.2.Ch. 6.12 - Prob. 6MPCh. 6.12 - Let F=2i3j+k act at the point (5,1,3). (a) Find...Ch. 6.12 - Prob. 8MPCh. 6.12 - Let F=i5j+2k act at the point (2,1,0). Find the...Ch. 6.12 - Given u=xy+sinz, find (a) the gradient of u at...Ch. 6.12 - Given =z23xy, find (a) grad ; (b) the directional...Ch. 6.12 - Given u=xy+yz+zsinx, find (a) u at (0,1,2); (b)...Ch. 6.12 - Given =x2yz and the point P(3,4,1), find (a) at...Ch. 6.12 - If the temperature is T=x2xy+z2, find (a) the...Ch. 6.12 - Show that...Ch. 6.12 - Given F1=2xzi+yj+x2k and F2=yixj: (a) Which F, if...Ch. 6.12 - Find the value of Fdr along the circle x2+y2=2...Ch. 6.12 - Is F=yi+xzj+zk conservative? Evaluate Fdr from...Ch. 6.12 - Given F1=2yi+(z2x)j+(y+z)k,F2=yi+2xj: (a) Is F1...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...
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