Partial Derivatives

Partial derivatives occur when taking the derivative of a function of two or more variables, e.g., $f(x,y)$.  If we want to determine the derivative of the function $f(x,y)$ with respect to one of the independent variables, we do this while holding the other independent variable(s) constant. This process is partial differentiation and is given in calculus terms as:

$$ \frac{\partial f(x,y)}{\partial x} = \lim_{x\rightarrow x_{o}} \frac{f\left(x,y_{c}\right)-f\left(x_{o},y_{c}\right)}{x-x_{o}}.$$

As mentioned above, we keep $y\rightarrow y_{c}$ constant while taking the derivative of the function with respect to $x$. Higher order partial derivatives are similar to normal derivatives and are commutative, for example:

$$\frac{\partial}{\partial x} \frac{\partial f(x,y)}{\partial y} = \frac{\partial}{\partial y} \frac{\partial f(x,y)}{\partial x}.$$

In cases where we are taking partial derivatives of a function dependent on another function(s), for example:

$$ f(g,h)=f(g(x,y),h(x,y),$$

then the partial derivatives of $f$ with respect to $x$ and $y$ are given by the chain rule, so that:

$$ \frac{\partial f(x,y)}{\partial x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x} + \frac{\partial f}{\partial h} \frac{\partial h}{\partial x}$$

and in similar fashion for the partial derivative with respect $y$. A common application of partial derivatives is found in vector calculus with the grad, div, and curl operators. Mathematically they are written as:

$$ \text{grad}\, f(x,y,z) = \nabla f = \frac{\partial f}{\partial x} \mathbf{a_x} + \frac{\partial f}{\partial y} \mathbf{a_y} + \frac{\partial f}{\partial z} \mathbf{a_z} $$

$$ \text{div}\, \mathbf{f}(x,y,z) = \nabla \cdot \mathbf{f} = \frac{\partial f_{x}}{\partial x} + \frac{\partial f_{y}}{\partial y}  + \frac{\partial f_{z}}{\partial z} $$

$$ \text{curl}\, \mathbf{f}(x,y,z) = \nabla \times \mathbf{f} = \frac{\partial f_{z}}{\partial y}-\frac{\partial f_{y}}{\partial z} \mathbf{a_x}+\frac{\partial f_{x}}{\partial z}-\frac{\partial f_{z}}{\partial x} \mathbf{a_y}+\frac{\partial f_{y}}{\partial x}-\frac{\partial f_{x}}{\partial y} \mathbf{a_z}$$

where $\mathbf{a_{i}}$ is the unit vector in Euclidean space. There is not much more to performing partial derivatives, but they are ubiquitous in partial differential equations and vector calculus which are essential mathematics for most engineers and scientists.

The selected quote comes from Alan Turing, the father of modern computer science.

"Science is a differential equation. Religion is a boundary condition."
-Alan Turing, 1954

References  & Additional Reading

[1] Advanced Engineering Mathematics, Wiley, 8th ed., E. Kreyszig.

[2] The Mathematics Companion, IoP, 1st ed., A.C. Fischer-Cripps.

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