Timeline for Why is negative divergence an adjoint of gradient?
Current License: CC BY-SA 3.0
11 events
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| Nov 8, 2017 at 5:24 | comment | added | user10024395 | @MichaelHardy Thanks for edit. | |
| Nov 7, 2017 at 17:54 | comment | added | Michael Hardy | $$ \begin{align} & \left( F_1 \frac{\partial f}{\partial x} + f\frac{\partial F_1}{\partial x} \right) + \left( F_2 \frac{\partial f}{\partial y} + f\frac{\partial F_2}{\partial y} \right) + \left( F_3 \frac{\partial f}{\partial z} + f\frac{\partial F_3}{\partial z} \right) \\ \\ = {} & \frac \partial {\partial x} (F_1 f) + \frac \partial {\partial y} (F_2 f) + \frac \partial {\partial z} (F_3 f) \qquad \text{by the product rule} \end{align} $$ | |
| Nov 7, 2017 at 17:53 | comment | added | Michael Hardy | $$ \begin{align} & F_1 \frac{\partial f}{\partial x} + F_2 \frac{\partial f}{\partial y} + F_3\frac{\partial f}{\partial z} \\ \\ \overset{\Large\text{?}}= {} & -f\frac{\partial F_1}{\partial x} -f\frac{\partial F_2}{\partial y} - f\frac{\partial F_3}{\partial z} \end{align} $$ | |
| Nov 7, 2017 at 17:53 | comment | added | Michael Hardy | $$\begin{align} F(x,y,z) = {} & (F_1(x,y,z), F_2(x,y,z), F_3(x,y,z)) \\ \\ \operatorname{div} F(x,y,z) = {} & \frac{\partial F_1}{\partial x} + \frac{\partial F_1}{\partial y} + \frac{\partial F_1}{\partial z} \\ \\ \nabla f(x,y,z) = {} & \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}, \right) \end{align}$$ | |
| Nov 7, 2017 at 17:41 | answer | added | Robert Lewis | timeline score: 5 | |
| Nov 7, 2017 at 17:27 | answer | added | Overflowian | timeline score: 1 | |
| Nov 7, 2017 at 17:10 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| Nov 7, 2017 at 17:09 | history | edited | user10024395 | CC BY-SA 3.0 |
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| Nov 7, 2017 at 17:06 | comment | added | Michael Hardy | See my edits to this question for proper MathJax usage. | |
| Nov 7, 2017 at 17:05 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| Nov 7, 2017 at 16:56 | history | asked | user10024395 | CC BY-SA 3.0 |