CHALLENGE: learning partial derivatives. The formal method to change coordinate systems from (ux, Uy, Uz), which I have noted in the formula below as (x,y,z) so that it is easier to read, to (u, 0, Ф ) involves evaluation of the Jacobian determinant J: where J = ?х ди ду [[[ h(x, y, z)dxdydz = [[ h(x(u, 0, Ф), y(u, 0, Ф), z(u, 0, Ф)\\dud0d ?x ?x дө аф ду ду дx (ду дz ?и до д ди дө аф дz дz дz ди де аф ду дz ?x (ду дz ?0 ди д ду дz до ди + дх ду дz ?ф?u де ду дz. де ди Solve for the Jacobian J for the coordinate transformation from Cartesian to spherical coordinates and rewrite the expression for F(u)du. Show that by integrating over the angular components of the entire sphere, the final expression for F(u)du is the same as that derived above.
CHALLENGE: learning partial derivatives. The formal method to change coordinate systems from (ux, Uy, Uz), which I have noted in the formula below as (x,y,z) so that it is easier to read, to (u, 0, Ф ) involves evaluation of the Jacobian determinant J: where J = ?х ди ду [[[ h(x, y, z)dxdydz = [[ h(x(u, 0, Ф), y(u, 0, Ф), z(u, 0, Ф)\\dud0d ?x ?x дө аф ду ду дx (ду дz ?и до д ди дө аф дz дz дz ди де аф ду дz ?x (ду дz ?0 ди д ду дz до ди + дх ду дz ?ф?u де ду дz. де ди Solve for the Jacobian J for the coordinate transformation from Cartesian to spherical coordinates and rewrite the expression for F(u)du. Show that by integrating over the angular components of the entire sphere, the final expression for F(u)du is the same as that derived above.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:c) CHALLENGE: learning partial derivatives. The formal method to change coordinate systems from (ux, Uy, uz),
which I have noted in the formula below as (x,y,z) so that it is easier to read, to (u, 0, Ф ) involves evaluation
of the Jacobian determinant J:
where J
-
?x
ди
ду
ди
дz
ди
[[[ h(x,y,z)dxdydz = [[| h(x(u,0, Ф), y(u,0, Ф), z(u,0, Ф)U\dud9d
?x
де
ду
?x
аф
ду
дө
аф
дz дz
дө аф
дx (ду дz
?u де д
ду дz
аф де
?x (dy dz
деди д
ду дz
до ди
+
?x ду дz
?ф?u де
ду дz,
де ?u
Solve for the Jacobian J for the coordinate tra sformation from Cartesian to spherical coordinates and
rewrite the expression for F(u)du. Show that by integrating over the angular components of the entire
sphere, the final expression for F(u)du is the same as that derived above.
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