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Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$

then the potential function (if the field is conservative) can be found by integrating $A$ with respect to $x$ and $B$ with respect to $y$ and then "unifying" the two solutions into a single function.

But what does the potential function really mean? Can somebody give me a simple intuitive explanation (maybe graphic) of what this function means?

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Consider a surface $z=F(x,y)$ in three-dimensional space. Suppose I have a massive point particle sitting on the surface at $(x,y)$, and uniform gravity acts downwards. What is the force felt by the particle, or what is the instantaneous acceleration going to be? Newton (and his successors) tell us that the particle will start to move towards the direction where the function $F$ decreases fastest. We find this by taking the gradient of $F$ at $(x,y)$, $\nabla F$. This is a conservative vector field (curl is zero, or use the fundamental theorem of calculus on the line integral).

(In other words: the potential is to the vector field as the gravitational potential energy is to the gravitational force field.)

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  • $\begingroup$ Thanks for your reply. So basically the potential function gives me the "steepness" at each point on the surface? $\endgroup$ – qmd May 9 '15 at 15:56
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    $\begingroup$ The potential function is the surface. The vector field points in the direction of greatest slope at each point of the surface. $\endgroup$ – Chappers May 9 '15 at 17:18
  • $\begingroup$ Ahhh! Now I get it. Thanks a lot :) Have a nice day $\endgroup$ – qmd May 10 '15 at 11:57

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