Question #2926

Find the divergence of the vector A=3x ux + xy[sup]2[/sup] uy - 2xy e[sup]-z[/sup] uz

Expert's answer

<img src="/cgi-bin/mimetex.cgi?A=%203x%20ux%20+%20xy%5E2%20uy%20-%202xy%20e%5E%7B-z%7D%20uz%20=3ux%5E2%20+%20uxy%5E3%20-%202uxyze%5E%7B-z%7D%20%5C%5C%0A%5Cfrac%7B%5Cpartial%7BA%7D%7D%7B%5Cpartial%7Bx%7D%7D%20=%206ux%20+%20uy%5E3%20-2uyze%5E%7B-z%7D%20%5C%5C%0A%5Cfrac%7B%5Cpartial%7BA%7D%7D%7B%5Cpartial%7By%7D%7D%20=%203uxy%5E2%20-2uxze%5E%7B-z%7D%20%5C%5C%0A%5Cfrac%7B%5Cpartial%7BA%7D%7D%7B%5Cpartial%7Bz%7D%7D%20=%20-2uxy%28e%5E%7B-z%7D%20-ze%5E%7B-z%7D%29%20%5C%5C%0Adiv%20A%20=%20%5Cfrac%7B%5Cpartial%7BA%7D%7D%7B%5Cpartial%7Bx%7D%7D+%5Cfrac%7B%5Cpartial%7BA%7D%7D%7B%5Cpartial%7By%7D%7D+%20%5Cfrac%7B%5Cpartial%7BA%7D%7D%7B%5Cpartial%7Bz%7D%7D%20%5C%5C" title="A= 3x ux + xy^2 uy - 2xy e^{-z} uz =3ux^2 + uxy^3 - 2uxyze^{-z} \\

\frac{\partial{A}}{\partial{x}} = 6ux + uy^3 -2uyze^{-z} \\

\frac{\partial{A}}{\partial{y}} = 3uxy^2 -2uxze^{-z} \\

\frac{\partial{A}}{\partial{z}} = -2uxy(e^{-z} -ze^{-z}) \\

div A = \frac{\partial{A}}{\partial{x}}+\frac{\partial{A}}{\partial{y}}+ \frac{\partial{A}}{\partial{z}} ">

\frac{\partial{A}}{\partial{x}} = 6ux + uy^3 -2uyze^{-z} \\

\frac{\partial{A}}{\partial{y}} = 3uxy^2 -2uxze^{-z} \\

\frac{\partial{A}}{\partial{z}} = -2uxy(e^{-z} -ze^{-z}) \\

div A = \frac{\partial{A}}{\partial{x}}+\frac{\partial{A}}{\partial{y}}+ \frac{\partial{A}}{\partial{z}} ">

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