Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Question
Chapter 10.9, Problem 14P
To determine
To fid:The scale factors
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
What are the magnitudes and directions of each of the following position vectors? When reporting a direction, make sure you give the amount of rotation, the direction of rotation, and the starting point for the rotation.
a. Ax =12.0m, Ay =16.0m
b. Bx =275m, By =−185m
c. Cx =−3.40m, Cy =2.60m
Find the magnitude and direction of the following vectors (illustrate them graphically).
v (0.2316, 0.4179)u (17.3, 78.4)
Consider the following scenario.
A pilot is steering a plane in the direction N 45° W at an air-speed (speed still in air) of 150 mi/h. A wind is blowing in the direction S 30° E at a speed of 34 mi/h.
Set up the coordinate axes so that north is the positive y-direction and west is the negative x-direction. With respect to the still air, write a vector that represents the velocity of the plane and a vector that represents the velocity of the wind.
Chapter 10 Solutions
Mathematical Methods in the Physical Sciences
Ch. 10.2 - Verify equations (2.6).Ch. 10.2 - Prob. 2PCh. 10.2 - Consider the matrix A in (2.7) or (2.10). Think of...Ch. 10.2 - Any rotation of axes in three dimensions can be...Ch. 10.2 - Write equations (2.12) out in detail and solve the...Ch. 10.2 - Write the transformation equation for a 3rd-rank...Ch. 10.2 - Following what we did in equations (2.14) to...Ch. 10.2 - Write the equations in (2.16) and so in (2.17)...Ch. 10.3 - Write equations (2.11,), (2.12), (2.13), (2.14),...Ch. 10.3 - Show that the fourth expression in (3.1) is equal...
Ch. 10.3 - As we did in (3.3), show that the contracted...Ch. 10.3 - Show that the contracted tensor TijkVk is a 2nd...Ch. 10.3 - Show that TijklmSlm is a tensor and find its rank...Ch. 10.3 - Show that the sum of two 3rd -rank tensors is a...Ch. 10.3 - As in problem 6, show that the sum of two 2nd...Ch. 10.3 - Show that (3.9) follows from (3.8). Hint: Give a...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Show that the first parenthesis in (3.5) is a...Ch. 10.4 - As in (4.3) and (4.4), find the y and z components...Ch. 10.4 - Complete Example 4 to verify the rest of the...Ch. 10.4 - As in Problem 2, complete Example 5.Ch. 10.4 - Find the inertia tensor about the origin for a...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.5 - Verify that (5.5) agrees with a Laplace...Ch. 10.5 - Verify for a few representative cases that (5.6)...Ch. 10.5 - Show that ijklm is an isotropic tensor of rank...Ch. 10.5 - Generalize Problem 3 to see that the direct...Ch. 10.5 - Let Tjkmn be the tensor in (5.8). This is a...Ch. 10.5 - Evaluate: (a) ijjkkmim (b) ijkjk (c) jk2k2j (d)...Ch. 10.5 - Write in terms of s as in (5.8) and (5.9): (a)...Ch. 10.5 - Show that the equations (5.10) are correct. Hints:...Ch. 10.5 - (a) Finish the work of showing that the cross...Ch. 10.5 - (a) Write the triple scalar product A(BC) in...Ch. 10.5 - Using problem 10, write A(BA) in tensor notation...Ch. 10.5 - Write and prove in tensor notation: (a) Chapter 6,...Ch. 10.5 - Write in tensor notation and prove the following...Ch. 10.5 - Show that the diagonal elements of an...Ch. 10.5 - Write a 4-by-4 antisymmetric matrix to show that...Ch. 10.5 - Verify that (5.16) gives (5.17). Also verify that...Ch. 10.5 - Write out the components of Tjk=AjBkAkBj to show...Ch. 10.6 - Show that in 2 dimension (say the x, y plane), an...Ch. 10.6 - In Chapter 3, we said that any 3-by-3 orthogonal...Ch. 10.6 - For Example 1, write out the components of U,V,...Ch. 10.6 - Do Example 1 and Problem 3 if the transformation...Ch. 10.6 - Write the tensor transformation equations for...Ch. 10.6 - Prob. 6PCh. 10.6 - Write the transformation equations for the triple...Ch. 10.6 - Write the transformation equations for WS to...Ch. 10.6 - Prob. 9PCh. 10.6 - Prob. 10PCh. 10.6 - Prob. 11PCh. 10.6 - Prob. 12PCh. 10.6 - Prob. 13PCh. 10.6 - Prob. 14PCh. 10.6 - In equation (5.12), find whether A(BC) is a vector...Ch. 10.6 - In equation (5.14), is (V) a vector or a...Ch. 10.6 - In equation (5.16), show that if Tjk is a tensor...Ch. 10.7 - Verify (7.1).Hints: In Figure 7.1, consider the...Ch. 10.7 - Write out the sums Pijej for each value of i and...Ch. 10.7 - Carry through the details of getting (7.4) from...Ch. 10.7 - Interpret the elements of the matrices in Chapter...Ch. 10.7 - Show by the quotient rule (Section 3) that Cijkm...Ch. 10.7 - If P and S are 2nd-rank tensors, show that 92=81...Ch. 10.7 - In (7.9) we have written the first row of elements...Ch. 10.7 - Do Problem 4.8 in tensor notation and compare the...Ch. 10.8 - Find ds2 in spherical coordinates by the method...Ch. 10.8 - Observe that a simpler way to find the velocity...Ch. 10.8 - Prob. 3PCh. 10.8 - In the text and problems so far, we have found the...Ch. 10.8 - Prob. 5PCh. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - Sketch or computer plot the coordinate surfaces in...Ch. 10.8 - Prob. 11PCh. 10.8 - Using the expression you have found for ds, and...Ch. 10.8 - Prob. 13PCh. 10.8 - Using the expression you have found for ds, and...Ch. 10.8 - Let x=u+v,y=v. Find ds, thea vectors, and ds2 for...Ch. 10.9 - Prove (9.4) in the following way. Using (9.2) with...Ch. 10.9 - Prob. 2PCh. 10.9 - Using cylindrical coordinates write the Lagrange...Ch. 10.9 - Prob. 4PCh. 10.9 - Write out U,V,2U, and V in spherical coordinates.Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Prob. 14PCh. 10.9 - Prob. 15PCh. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.18) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8) and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.10 - Verify equation (10.7). Hint: Use equations (2.4)...Ch. 10.10 - From (10.1) find /x=(1/r)coscos and show that...Ch. 10.10 - Divide equation (10.4) by dt to show that the...Ch. 10.10 - Prob. 4PCh. 10.10 - Write u in polar coordinates in terms of its...Ch. 10.10 - Prob. 6PCh. 10.10 - As in (10.12), write the transformation equations...Ch. 10.10 - Using (10.15) show that gij is a 2nd-rank...Ch. 10.10 - If Ui is a contravariant vector and Vj is a...Ch. 10.10 - Show that if Vi is a contravariant vector then...Ch. 10.10 - In (10.18), show by raising and lowering indices...Ch. 10.10 - Show that in a general coordinate system with...Ch. 10.10 - Verify (10.20).Ch. 10.10 - Using equations (10.20) to (10.23), write the...Ch. 10.10 - Do Problem 14 for an orthogonal coordinate system...Ch. 10.10 - Continue Problem 8.15 to find the gij matrix and...Ch. 10.10 - Repeat Problems 8.15 and 10.16 above for the (u,v)...Ch. 10.10 - Using (10.19), show that aiai=ji.Ch. 10.11 - Show that the transformation equation for a...Ch. 10.11 - Let e1,e2,e3 be a set of orthogonal unit vectors...Ch. 10.11 - In Chapter 3, Problem 6.6, you are asked to prove...Ch. 10.11 - If E= electric field and B= magnetic field, is EB...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - If u is a vector specifying the displacement under...Ch. 10.11 - Show that elements Rij of a rotation matrix are...Ch. 10.11 - Show that the nine quantities Tij=Vi/xj (which are...Ch. 10.11 - The square matrix in equation (10.3) is called the...Ch. 10.11 - In equation (10.13) let the x variables be...
Knowledge Booster
Similar questions
- For the vector v=(2,1), sketch a 2v, b 3v, and c 12v.arrow_forwardConsider the following scenario. A pilot is steering a plane in the direction N 45° W at an air-speed (speed still in air) of 190 mi/h. A wind is blowing in the direction S 30° E at a speed of 34 mi/h. Set up the coordinate axes so that north is the positive y-direction and west is the negative x-direction. With respect to the still air, write a vector that represents the velocity of the plane and represents the velocity of the wind. Vplane = Vwind= Give a velocity vector of the plane relative to ground. (Round your values to three decimal places.) V = Vplane + wind= Find vl. (Round your answer to one decimal place.) |v| = Find the true course and the ground speed (in mi/h) of the plane. (Round your answers to one decimal place.) true course N W ground speed Need Help? Read It mi/h vector thatarrow_forwardProb. 4. Find the angle between two vectors i = (1,–1,0), and i = (2,-1,2)arrow_forward
- Vectors are quantities that possess both magnitude and direction. In engineering problems, it is best to think of vectors as arrows, and usually it is best to manipulate vectors using components. In this tutorial, we consider the addition of two vectors using both of these techniques. Consider two vectors A and B that have lengths A and B, respectively. Vector B makes an angle from the direction of A. (Figure 1)In vector notation, the sum is represented by C = A +B where C is a new vector that is the sum of A and B. b A с a Barrow_forwardIn the figure to the right, it looks as if v1+v2 and v1−v2 are orthogonal. Under what circumstances would we expect the sum of two vectors to be orthogonal to their difference? Choose the correct answer below. They will be orthogonal if v1 and v2 do not have the same direction. They will be orthogonal if v1 and v2 are orthogonal. They will be orthogonal if v1 and v2 have the same magnitude. They will be orthogonal if either v1 or v2 is a unit vector.arrow_forwardQuestion 3 3.1 An airplane heads due north at 650 km/h. It experiences a 60 km/h crosswind flowing in the direction N 30°W 3.1.1 Express the velocity v of the airplane relative to the air in component form. 3.1.2 Express the velocity w of the wind in component form. 3.1.3 Find the true velocity of the airplane as a velocity of the airplane as a vector. 3.1.4 Find the true speed and direction of the airplane.arrow_forward
- Q.6arrow_forwardQ.2 Find the vector whose magnitude is 5 and which is in the direction of the vector 4i – 3j + k.arrow_forwardVector A is 3 units long at 25° and vector B is 3.5 units long at 105°. Draw the two vectors in a coordinate system schematically. Draw the sum of the two vectors in the same diagram. Calculate the x and y components of each vector. Calculate the x and y components of the vector C=A+B using the component method. .......................At time t=0 s, a stone is thrown from the top of a building horizontally with initial velocity 24 m/s. The building is so tall that the stone is still in the air at t=1.5 s. Calculate The horizontal displacement of the stone at 1=1.5 s. The vertical displacement of the stone at t=1.5 sarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning