Essentials of Math Methods for Physicists by Hans J. Weber and George B. Arfken (Auth.)

By Hans J. Weber and George B. Arfken (Auth.)

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5 GRADIENT, V PARTIAL DERIVATIVES In this Section we deal with derivatives of functions of several variables which will lead us to the concept of directional derivative or gradient operator that is of central importance in mechanics, electrodynamics and engineering. 5. GRADIENT, 37 V We can view a function z =

Net rate of flow out| x = ~^-(pvx) dx dy dz. Equivalently, we can arrive at this result by lim pvx(Ax,0i0)-pvx Δζ->0 (0,0,0) _ d[pvx{x,y,z)) dx Δχ ο,ο,ο Now the x-axis is not entitled to any preferred treatment. /-axis, with x replaced by y and the corresponding changes for y and z\ y —> z, z —► x. This is a cyclic permutation of the coordinates. A further cyclic permutation yields the result for the remaining two faces of our parallelepiped. Adding the net rate of flow out for all three pairs of surfaces of our volume element, we have net flow out (per unit time) Υχ = V · (pvx) + ^ (pvy) + (pv)dxdydz.

How do Fi and F2 compare? (c) Calculate Fi and F2 for the case of q\ and q2 moving along parallel trajectories side by side. ANS. 5 GRADIENT, V PARTIAL DERIVATIVES In this Section we deal with derivatives of functions of several variables which will lead us to the concept of directional derivative or gradient operator that is of central importance in mechanics, electrodynamics and engineering. 5. GRADIENT, 37 V We can view a function z =

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