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Showing posts from April, 2019

Visual calculus

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I've been wondering: is it possible to visualize the rules of calculus? 1. Integration by parts To simplify things, in the above visualization v(x_1) = 0, so that \int^{x'}_{x_1}\frac{dv}{dx}dx = v(x').  This means that v(x_2) is simply the area of the cross-section facing us on the LHS, and the volume  on the LHS is u(x_2)v(x_2). The RHS shows the same volume split into two parts.  The rectangle embedded in the first is u\frac{dv}{dx}, and so its volume is \int^{x_2}_{x_1}u\frac{dv}{dx}dx.  The cross section of the 2nd part is v(x), so its volume is \int^{u_2}_{u_1}vdu, which becomes \int^{x_2}_{x_1}v\frac{du}{dx}dx when rewritten as an integral over x. So the picture is visual proof that the equation in pink holds, provided that v(x_1) = 0.  To prove it in general we just need to check that when we replace v with v+v_1 in the equation, both sides change by the same amount. 2. The chain rule This picture demonstr...