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By K. F. Riley
This scholar answer guide offers whole ideas to the entire odd-numbered difficulties in crucial Mathematical tools for the actual Sciences. It takes scholars via every one challenge step by step, to allow them to sincerely see how the answer is reached, and comprehend any error of their personal operating. scholars will examine by way of instance the right way to decide upon a suitable process, enhancing their problem-solving talents.
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Additional resources for Essential Mathematical Methods for the Physical Sciences: Student Solution Manual
Example text
Given that B(0) = 0 and that B is continuous at ρ = a, obtain expressions for C and B(ρ) in terms of B0 and a. 35 Vector calculus (b) The magnetic field can be expressed as B = ∇ × A, where A is known as the vector potential. Show that a suitable A can be found which has only one non-vanishing component, Aφ (ρ), and obtain explicit expressions for Aφ (ρ) for both ρ < a and ρ > a. Like B, the vector potential is continuous at ρ = a. (c) The gas pressure p(ρ) satisfies the hydrostatic equation ∇p = J × B and vanishes at the outer wall of the cylinder.
Thus, zB F = 2πρ0 g ρz(−dρ) zA = −2πρ0 g zB ρ zA = −2πρ0 g z ρ2 2 ∂ρ z dz ∂z zB − zA zB zA ρ2 dz . 2 But ρ(zA ) = ρ(zB ) = 0, and so the first contribution vanishes, leaving F = ρ0 g zB πρ 2 dz = ρ0 gV , zA where V is the volume of the solid. This is the mathematical form of Archimedes’ principle. Of course, the result is also valid for a closed body of arbitrary shape, ρ = ρ(z, φ), but a different method would be needed to prove it. 13 A vector field a is given by −zxr −3 i − zyr −3 j + (x 2 + y 2 )r −3 k, where r 2 = x 2 + y 2 + z2 .
Direct evaluation of all the separate cases is as follows: (x1 )† Ax2 = 1 + 0 − 1 = 0, (x1 )† Ax3 = 1 − 2 + 1 = 0, (x2 )† Ax3 = 1 + 0 − 1 = 0, (x1 )† Bx2 = (x1 )† (1 0 −1)T = 1 + 0 − 1 = 0, (x1 )† Bx3 = (x1 )† (2 − 4 2)T = 2 − 4 + 2 = 0, (x2 )† Bx3 = (x2 )† (2 − 4 2)T = 2 + 0 − 2 = 0. If (xi )† Axj has zero value then so does (xj )† Axi (and similarly for B). So there is no need to investigate the other six possibilities and the verification is complete. (b) In order to determine the behavior of the system we need to know which modes are present in the initial configuration.