Partial Differentiation and its Applications

This post is going to introduce the concept of partial derivatives. Partial differentiation is similar to regular differentiation only involving more variables.

In the partial differentiation of functions involving multiple variables much of the same rules apply as in regular differentiation. We choose the one variable in which we differentiate with respect to and the other variable is considered as a constant. As such there are multiple differential equations each in respect to different variables.

For a function involving two variables we can define the partial derivatives as follows.

      \begin{align*}    f_x(x,y) = \lim_{h \to 0}{\frac{f(x+h,y) - f(x,y)}{h}} \\    f_y(x,y) = \lim_{h \to 0}{\frac{f(x,y+h) - f(x,y)}{h}} \end{align*}

For partial differentiation we use the following notation to distinguish it from regular differentiation.

      \begin{align*}    f_x(x,y) = \frac{ \partial }{ \partial x} f(x,y) = \frac{ \partial z }{ \partial x}  \\    f_y(x,y) = \frac{ \partial }{ \partial y} f(x,y) = \frac{ \partial z }{ \partial y}  \\ \end{align*}

One important application of partial derivatives is in finding the area of a curved surface. Before the discovery of calculus this was done by approximation. Archimedes is credited for finding the formulas for the surface areas of many 3-dimensional curved objects.

The following is the general equation for the surface integral of a 3-dimensional object in terms of z.

     \[    \int \int _S {f(x,y,z)} \,\mathrm{d}S  \] \[    \int \int _D {f(x,y,g(x,y)) \sqrt{   \left(  \frac{\partial z}{ \partial x} \right) ^ 2 +   \left( \frac{\partial z}{ \partial y} \right) ^ 2  + 1  } } \,\mathrm{d}A  \]

This equation, and partial differentiation in general, is incredibly useful in solving many problems in both mathematics and physics itself. In the next post, which will be on the subject of the paradox of Gabriel’s horn, I will make use of the above equation. Also in the future, once I’ve establish a strong mathematical foundation, I will begin to demonstrate the importance of calculus in quantum theory. The particular example of the use of partial differentiation that im refering to is in its importance in regards to the Heisenberg’s Uncertainty principal.

Dan Roy, Founder and Creator of Universal Physics.

Category(s): Mathematics
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5 Responses to Partial Differentiation and its Applications

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    this is mental says:

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    Leonard Marks says:

    great post

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