Verify the following relations for cylindrical coordinates u = r, v = 0, w = z: a) the surfaces r= const, 0 = const, z = const form a triply orthogonal family, and a(x, y, z) =r; a(r, 0, z) b) the element of are length is given by ds? = dr² +r²d0² + dz?; c) the components of a vector p are given by Pr = Px cos 0 + Py sin 0, Po = -Px sin 0 + Py cos 0, P: = Pz; d) grad U has components: 2U lâu au. e) divp= (rp.) + t); f) curl p has components: ape ap apr ae az az ar ae g) v²u is given by (3.66). v²U = au a2U + əz²
Verify the following relations for cylindrical coordinates u = r, v = 0, w = z: a) the surfaces r= const, 0 = const, z = const form a triply orthogonal family, and a(x, y, z) =r; a(r, 0, z) b) the element of are length is given by ds? = dr² +r²d0² + dz?; c) the components of a vector p are given by Pr = Px cos 0 + Py sin 0, Po = -Px sin 0 + Py cos 0, P: = Pz; d) grad U has components: 2U lâu au. e) divp= (rp.) + t); f) curl p has components: ape ap apr ae az az ar ae g) v²u is given by (3.66). v²U = au a2U + əz²
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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![Verify the following relations for cylindrical coordinates u = r, v = 0, w = z:
a) the surfaces r = const, 0 = const, z = const form a triply orthogonal family, and
a(x, y, z)
=r;
a(r, 0, z)
b) the element of are length is given by
ds? = dr² +r²d0² + dz?;
c) the components of a vector p are given by
P, = Px cos 0 + py sin 0,
Po = -Px sin 0 + Py cos 0, P: = P2i
d) grad U has components: aU làu aU.
e) divp= (rp,) + +r];
f) curl p has components:
ape
apr
apr
az
az
ar
ae
g) v²u is given by (3.66).
au
a?U
v²U =
ar
+
ar
a02
az?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1c4d217-886f-4958-b018-18e9293fbd17%2Ff7ada637-d33f-4810-af31-18e08fd51ae3%2Fh4r5kpb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Verify the following relations for cylindrical coordinates u = r, v = 0, w = z:
a) the surfaces r = const, 0 = const, z = const form a triply orthogonal family, and
a(x, y, z)
=r;
a(r, 0, z)
b) the element of are length is given by
ds? = dr² +r²d0² + dz?;
c) the components of a vector p are given by
P, = Px cos 0 + py sin 0,
Po = -Px sin 0 + Py cos 0, P: = P2i
d) grad U has components: aU làu aU.
e) divp= (rp,) + +r];
f) curl p has components:
ape
apr
apr
az
az
ar
ae
g) v²u is given by (3.66).
au
a?U
v²U =
ar
+
ar
a02
az?
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