How a Multivariable Function Changes with Respect to a Polar Coordinate

For a functions $f(x)$, $g(y)$ and $h(z)$, the chain rule yields $ \frac{\partial f(x)}{\partial \theta} = ( \frac{\partial f}{\partial x} ) ( \frac{\partial x}{\partial \theta} )$. $ \frac{\partial g(y)}{\partial \theta}  = (\frac{\partial g}{\partial y})…

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How a Multivariable Function Changes with Respect to a Radial Coordinate

First focus on the Cartesian coordinate $x$, which depends on the spherical coordinates $r,\theta,$ and $\phi$. For a function $f(x)$, the chain rule yields $ \frac{\partial f(x)}{\partial r} = ( \frac{\partial f}{\partial x} )…

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Indefinite Integration

The following quote from Reference [1] can clarify a lot of confusion about indefinite integration: Today the process of finding the fluent of a given fluxion is called indefinite integration, or antidifferentiation, and…

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Partial Fractions

This post is based on a section called Integration by Partial Fractions in Morris Kline’s book on Calculus [1]. The first step in understanding partial fractions is learning about polynomials. Definition of a Polynomial From…

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