Killian O'Brien
2014
We will cover three main areas
Recall that for a function, \(f: \mathbb{R} \to \mathbb{R}\), of a single variable the derivative, \(f'(a)\), of \(f\) at a point \(a\) is defined by \[f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}.\]
An equivalent formulation of the definition, using alterntaive notation, is \[ \frac{df}{dx} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}. \]
Let \(f\) be a function of the two independent variables \(x\) and \(y\). The partial derivative, \(\frac{ \partial f}{\partial x}\), of \(f\) with respect to \(x\) is defined as \[ \frac{\partial f}{\partial x} = \lim_{\delta x \to 0} \frac{f(x + \delta x, y) - f(x,y)}{\delta x}. \]
Similarly we can define the partial derivative of \(f\) with respect to the other variable. The partial derivative, \(\frac{ \partial f}{\partial y}\), of \(f\) with respect to \(y\) is defined as \[ \frac{\partial f}{\partial y} = \lim_{\delta y \to 0} \frac{f(x , y+ \delta y) - f(x,y)}{\delta y}. \]
Consider the function \(f\) defined by \[f(x,y) = x^2 y.\] The partial derivatives are evaluated as follows, \[\begin{align} \frac{\partial f}{\partial x} &= \lim_{\delta x \to 0} \frac{f(x + \delta x, y) - f(x,y)}{\delta x} ,\\ &= \lim_{\delta x \to 0} \frac{\left ( x + \delta x \right )^2 y - x^2 y}{\delta x} ,\\ &= \lim_{\delta x \to 0} \frac{2x \delta x + (\delta x)^2}{\delta x} y ,\\ &= \lim_{\delta x \to 0} ( 2x + \delta x ) y ,\\ &= 2xy. \end{align} \] A similar calculation will show that \[ \frac{ \partial f}{\partial y} = x^2. \]
In practice we make use of all the standard derivatives and properties of differentiation that we have from the one-variable case, along with the following principle:
when differentiating a multi-variable function, \(f\) say, with respect to the variable \(x\), we treat the other variables as if they were constants, so that \(f\) becomes a function of the single-variable \(x\), and then differentiate \(f\) using the standard one-variable differentiation techniques
Consider the function \(f\) defined by \[ f(x,y) = x^2 y^3 \tan{(2x)} ,\] and find its partial derivative \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).
>> syms x y
>> diff(x^2*y^3*tan(2*x),x)
ans =
x^2*y^3*(2*tan(2*x)^2 + 2) + 2*x*y^3*tan(2*x)The derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) can also be written \(f_x\) and \(f_y\) respectively. Makes it easier to refer to values of the drivative, e.g. \[ f_x (0.1,4.5) \text{ as opposed to } \left. \frac{\partial f}{\partial x} \right |_{(x,y)=(0.1,4.5)} .\]
However the delta notation lends itself to seeing differentiation as an operator applied to a function, i.e. \[ \frac{\partial }{\partial x} \left ( \cos(xy) \right ) = -y\sin(xy).\]
Let \((x,y,z)\) be the usual Cartesian coordinate system for \(\mathbb{R}^3\). The equation z = f(x,y), defines a surface in \(\mathbb{R}^3\). For each pair \((a,b)\), the value of \(z=f(a,b)\) gives the height of the surface above or below (accordingly as \(z\) is positive or negative) the \((x,y)\)-plane. An example is shown in figure 1.1 (pg. 3) When evaluated at the point \((a,b)\), the partial derivatives, \[ \left. \frac{\partial f}{\partial x} \right |_{(x,y)=(a,b)}, \text{ and } \left. \frac{\partial f}{\partial y} \right |_{(x,y)=(a,b)}, \] give the gradient of the surface at the point \((a,b)\) in the \(x\) and \(y\) directions respectively.
Revise ordinary (one variable) derivatives: derivatives of standard functions, properties of the derivative (linearity, product rule, quotient rule, chain rule, ... )
Schaum's Outlines of Calculus by Ayres & Mendelson: Chapter 48 Partial Derivatives
Notes: Read section 1.4 and begin Exercise 1.1 Qs. 1,2,3
Explore Matlab commands: diff, ezsurf, ezplot3, ...
Learning about the Sage software is not a requirement of the course. I will use it mainly for showing examples in lectures and verifying calculations. However if you are interested in mathematics on the computer and in programming then you should definately explore it.
www.sagemath.org and cloud.sagemath.org