I understand how to use the distance formula to arrive at this:

"An infinitesimal length in the rectangular system is given by $dL^2= dx^2+dy^2+dz^2$ ."

How do you prove the following?

"In the cylindrical system the corresponding quantities are $dL^2= dr^2+r^2dφ^2+dz^2$ ."


I think you mean to put $$dL^2=dx^2+dy^2+dz^2\\dL^2=dr^2+r^2dφ^2+dz^2$$ For your proof, in a cylindrical coordinate system $$ r^2=x^2+y^2\\x=rcos(φ)\\y=rsin(φ)\\z=z $$ and because of product rule $$ \\dx=d(rcos(φ))=dr*cos(φ)+r*d(cos(φ)) $$ and $$\frac{d}{dφ}cos(φ)=-sin(φ)\\d(cos(φ))=-sin(φ)dφ$$ it follows that $$\\dx=cos(φ)dr-rsin(φ)dφ $$

and just substitute that for dx and find what dy is in terms of r and φ.

The expression $$d(f(x)g(x))=f(x)*dg(x)+g(x)*df(x)$$

is just the product rule. If you multiply both sides of the equation by 1/dx then it looks much more familiar.

  • $\begingroup$ Yes, I edited and corrected the equations. Thanks. $\endgroup$ – Chris C Mar 31 '18 at 14:09

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