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Consider the bipolar coordinates (s, t, v), related to the cartesian coordinates by 3 sinh(t) 3 sin(s) X = z = U cosh(t) – cos(s) y = cosh(t) – cos(s) where 0 < s < 2n and -o < t, v < ∞. Assuming that the scale factors are given by 3 3 h, = hị cosh(t) – cos(s) h, = 1, cosh(t) – cos(s) use the general form of the Laplacian operator in general curvilinear coordinates to give the form of the Laplacian operator in the bipolar coordinates Af = ds? dv²

Consider the bipolar coordinates (s, t, v), related to the cartesian coordinates by 3 sinh(t) 3 sin(s) X = z = U cosh(t) – cos(s) y = cosh(t) – cos(s) where 0 < s < 2n and -o < t, v < ∞. Assuming that the scale factors are given by 3 3 h, = hị cosh(t) – cos(s) h, = 1, cosh(t) – cos(s) use the general form of the Laplacian operator in general curvilinear coordinates to give the form of the Laplacian operator in the bipolar coordinates Af = ds? dv²

Advanced Engineering Mathematics
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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Consider the bipolar coordinates (s, t, v), related to the cartesian coordinates by 3 sinh(t) 3 sin(s) X = z = v cosh(t) – cos(s)’ y = cosh(t) – cos(s)* where 0 < s < 2n and – < t, v < ∞. Assuming that the scale factors are given by 3 3 hs cosh(t) – cos(s)’ h; : cosh(t) – cos(s) h, = 1, use the general form of the Laplacian operator in general curvilinear coordinates to give the form of the Laplacian operator in the bipolar coordinates Af = +. ds2 dv2
Transcribed Image Text:Consider the bipolar coordinates (s, t, v), related to the cartesian coordinates by 3 sinh(t) 3 sin(s) X = z = v cosh(t) – cos(s)’ y = cosh(t) – cos(s)* where 0 < s < 2n and – < t, v < ∞. Assuming that the scale factors are given by 3 3 hs cosh(t) – cos(s)’ h; : cosh(t) – cos(s) h, = 1, use the general form of the Laplacian operator in general curvilinear coordinates to give the form of the Laplacian operator in the bipolar coordinates Af = +. ds2 dv2
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