Explanation Calculation: The spherical coordinates are x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ Then differentials are calculated as follows x = r sin θ cos ϕ d x = sin θ cos ϕ d r + r cos θ cos ϕ d θ − r sin θ sin ϕ d ϕ y = r sin θ sin ϕ d y = sin θ sin ϕ d r + r cos θ sin ϕ d θ + r sin θ cos ϕ d ϕ z = r cos θ d z = cos θ d r − r sin θ d θ d s 2 = d x 2 + d y 2 + d z 2 d x 2 = sin 2 θ cos 2 ϕ d r 2 + r 2 cos 2 θ cos 2 ϕ d θ 2 + r 2 sin 2 θ sin 2 ϕ d ϕ 2 Since this is orthogonal system, the products d r d θ , d r d ϕ , d θ d ϕ are zeroes d y 2 = sin 2 θ sin 2 ϕ d r 2 + r 2 cos 2 θ sin 2 ϕ d θ 2 + r 2 sin 2 θ cos 2 ϕ d ϕ 2 d z 2 = cos 2 θ d r 2 + r 2 sin 2 θ d θ 2 d s 2 = d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 d s 2 = ( d r d θ d ϕ ) ( 1 2 0 0 0 r 2 0 0 0 r 2 sin 2 θ ) ( d r d θ d ϕ ) If x, y, z are functions in variables x 1 , x 2 , x 3 is orthogonal system with scale factors h 1 , h 2 , h 3 then d s 2 = ( d x 1 d x 2 d x 3 ) ( h 1 2 0 0 0 h 2 2 0 0 0 h 3 2 ) ( d x 1 d x 2 d x 3 ) Compare with matrix product, the scale factors of spherical coordinate system is h r = 1 , h θ = r , h ϕ = r sin θ If base vectors are a r , a θ , a ϕ then d s vector can be written as follows d s = e 1 h 1 d x 1 + e 2 h 2 d x 2 + e 3 h 3 d x 3 d s = e r ( 1 ) d r + e θ ( r ) d θ + e ϕ ( r sin θ ) d ϕ d s = e r d r + e θ r d θ + e ϕ r sin θ d ϕ ......... ( 1 ) The volume element of this system is d V = h 1 h 2 h 3 d x 1 d x 2 d x 3 d V = h r h θ h ϕ d r d θ d ϕ d V = r 2 sin θ d r d θ d ϕ The d s vector can be written in ( i , j , k ) system as follows d s = i d x + j d y + k d z d s = i ( sin θ cos ϕ d r + r cos θ cos ϕ d θ − r sin θ sin ϕ d ϕ ) + j ( sin θ sin ϕ d r + r cos θ sin ϕ d θ + r sin θ cos ϕ d ϕ ) + k ( cos θ d r − r sin θ d θ ) d s = ( i sin θ cos ϕ + j sin θ sin ϕ + k cos θ ) d r + ( i r cos θ cos ϕ + j r cos θ sin ϕ − k r sin θ ) d θ + ( − i r sin θ sin ϕ + j r sin θ cos ϕ ) d ϕ d s = ( i sin θ cos ϕ + j sin θ sin ϕ + k cos θ ) d r + ( i cos θ cos ϕ + j cos θ sin ϕ − k sin θ ) r d θ + ( − i sin ϕ + j cos ϕ ) r sin θ d ϕ ........ ( 2 ) To find base vectors, compare both expressions (1) and (2) for element d s e r = i sin θ cos ϕ + j sin θ sin ϕ + k cos θ e r = i sin θ cos ϕ + j sin θ sin ϕ + k cos θ is unit vector. e θ = i cos θ cos ϕ + j cos θ sin ϕ − k sin θ is unit vector. e ϕ = − i sin ϕ + j c o s ϕ , e ϕ is unit vector Corresponding a vector is a r = e r a θ = r e θ = i r cos θ cos ϕ + j r cos θ sin ϕ − k r sin θ a ϕ = r sin θ e ϕ = − i r sin θ sin ϕ + j r sin θ cos ϕ To find velocity divide d s element with d t d s = e r d r + e θ r d θ + e ϕ r sin θ d ϕ d s d t = e r d r d t + e θ r d θ d t + e ϕ r sin θ d ϕ d t d s d t = e r r ˙ + e θ r θ ˙ + e ϕ r sin θ ϕ ˙ ............ ( 3 ) Consider the particle s at point ( x , y , z ) s = ( r sin θ cos ϕ ) i + ( r sin θ sin ϕ ) j + ( r cos θ ) k s = r ( sin θ cos ϕ i + sin θ sin ϕ j + cos θ k ) s = r e k The velocity of this particle is d d t ( s ) = d d t ( r e r ) d s d t = d r d t . ( e r ) + r . d d t ( e r ) d d t ( e r ) = d d t ( i sin θ cos ϕ + j sin θ sin ϕ + k cos θ ) = i cos θ cos ϕ d θ d t − i sin θ sin ϕ d ϕ d t + j cos θ sin ϕ d θ d t + j sin θ cos ϕ d ϕ d t − k sin θ d θ d t d d t ( e r ) = d θ d t ( i cos θ cos ϕ + j cos θ sin ϕ − k sin θ ) + d ϕ d t ( − i sin θ sin ϕ + j sin θ c o s ϕ ) d d t ( e r ) = d θ d t e θ + d ϕ d t sin θ e ϕ d d t ( e r ) = θ ˙ e θ + ϕ ˙ sin θ e ϕ The velocity vector of particle is d s d t = d r d t

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Chapter 10.8, Problem 3P

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The velocity and acceleration components in spherical coordinates.

Also find the velocity starting with ds and with s=rer

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Chapter 10.8, Problem 2P

Chapter 10.8, Problem 4P

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Mathematical Methods in the Physical Sciences

Ch. 10.2 - Verify equations (2.6).Ch. 10.2 - Prob. 2P Ch. 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. 6P Ch. 10.6 - Write the transformation equations for the triple...Ch. 10.6 - Write the transformation equations for WS to...Ch. 10.6 - Prob. 9P Ch. 10.6 - Prob. 10P Ch. 10.6 - Prob. 11P Ch. 10.6 - Prob. 12P Ch. 10.6 - Prob. 13P Ch. 10.6 - Prob. 14P Ch. 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. 3P Ch. 10.8 - In the text and problems so far, we have found the...Ch. 10.8 - Prob. 5P 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 - 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. 11P Ch. 10.8 - Using the expression you have found for ds, and...Ch. 10.8 - Prob. 13P Ch. 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. 2P Ch. 10.9 - Using cylindrical coordinates write the Lagrange...Ch. 10.9 - Prob. 4P Ch. 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. 14P Ch. 10.9 - Prob. 15P 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 (9.18) 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. 4P Ch. 10.10 - Write u in polar coordinates in terms of its...Ch. 10.10 - Prob. 6P Ch. 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...

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The velocity and acceleration components in spherical coordinates. Also find the velocity starting with d s and with s = r e r

Solution Summary: The author explains the velocity and acceleration components in spherical coordinates.

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To find:

The velocity and acceleration components in spherical coordinates.

Also find the velocity starting with ds and with s=rer

Want to see the full answer?

Chapter 10 Solutions

The velocity and acceleration components in spherical coordinates. Also find the velocity starting with d s and with s = r e r

Solution Summary: The author explains the velocity and acceleration components in spherical coordinates.

Concept explainers

To find: The velocity and acceleration components in spherical coordinates. Also find the velocity starting with ds and with s=rer

Want to see the full answer?

Chapter 10 Solutions

To find:

The velocity and acceleration components in spherical coordinates.

Also find the velocity starting with ds and with s=rer