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1) They are slightly different in the sense that the function $C(x)$ (or $\vec C(\vec r)$) has different values at different places in space, and so even though the two opposite faces of the cube are very close the value of the function $C$ at the two faces will be slightly different. 2) Only if the two points are infinitesimally close or the function $C$ ...


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The minus sign is wrong.The reason for this is the x which you have chosen to be positive but is in fact negative. x points positively to the right and negatively to the left,and the horizontal vector that you are using in your picture is opposite to the direction of x.So,its x=-acotĪø. Cheers!


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This is the general form in which a conservation law can be expressed, when the quantity being conserved can't be converted in other forms, or disappear and reappear anywhere else. The flux density $\mathbf h$ is defined as that vector field that satisfies the equation you wrote, for every closed surface. More precisely, $\mathbf h \cdot \mathbf n$ ...


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It's just conservation of energy without being called as such. $$\mbox{Energy}_{\mbox{before}}=\mbox{Energy}_{\mbox{after}}$$ $$\frac{1}{2}m v_1^2+m g h_1=\frac{1}{2}m v_2^2 +m g h_2$$ $$\frac{1}{2}m v_1^2=\frac{1}{2}m v_2^2 +m g (h_2-h_1)$$ $$v_1^2=v_2^2 +2g \Delta y$$ This seems trivial when you have calculus and a concept of energy, but without ...


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You should be able to use it to solve constrained optimization. Let me give you sort of the big-picture overview. So you know that if $du$ is small compared to $u$ then you can linearize about $u$. When you want to generalize to more variables, you have to do partial derivatives: for example $$ f(x + dx, y + dy, z + dz) \approx f(x,y,z) + ...



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