Multivariable Functions Notes

Notes for Multivariable Functions

Functions of more than one variable are frequently encountered in scientific applications, when thinking about quantities which vary in more than one direction in space, or which vary in space and time.

Functions of two variables:

Functions of two variables have the form f(x,y). A new variable z can be introduced to visualise the function as a surface in the xyz plane, such that z=f(x,y). The variables x and y are independent, thus the behaviour of f in any direction can be formed from separate components in the x and y directions.

Partial derivatives:

Partial derivatives represent the rate of change of a multivariable function with respect to one of its variables. Geometrically, this corresponds to the function’s rate of change in the direction of one of its basis vectors. For a two-variable function:

Where $\left( \frac{\partial f}{\partial x} \right)_{y}$ is obtained by differentiating f with respect to x with y held constant, and $\left( \frac{\partial f}{\partial y} \right)_{x}$ is obtained by differentiating f with respect to y with x held constant.

The gradient vector (f):

At a given point in the plane defined by z=f(x,y), one can travel in any direction relative to the x and y axes. Thus, the gradient varies with direction – it is a directional, vector quantity. It is denoted grad(f) or f, where is the vector differential operator, defined in n dimensions as:

$$\boxed{\mathbf{\nabla}\mathbf{=}\left( \frac{\mathbf{\partial}}{\mathbf{\partial}\mathbf{x}_{\mathbf{1}}}\mathbf{,}\frac{\mathbf{\partial}}{\mathbf{\partial}\mathbf{x}_{\mathbf{2}}}\mathbf{,}\frac{\mathbf{\partial}}{\mathbf{\partial}\mathbf{x}_{\mathbf{3}}}\mathbf{,\ldots,}\frac{\mathbf{\partial}}{\mathbf{\partial}\mathbf{x}_{\mathbf{n}}} \right)}$$

Directional derivatives:

The gradient vector f points in the direction of steepest ascent on the surface f. The gradient in any direction at a given point (x,y) can be obtained by taking the projection of f onto a unit vector in that direction, using the scalar product. Such a derivative is called a directional derivative, and is defined by:

$$\boxed{\nabla f \bullet \widehat{\mathbf{u}}}$$

Where $\widehat{\mathbf{u}}$ is the unit vector in the appropriate direction.

Second and higher-order partial derivatives:

Higher-order partial derivatives are obtained in the same way as higher-order ordinary derivatives; by repeated differentiation:

The derivatives $\frac{\partial^{2}f}{\partial x\partial y}$ and $\frac{\partial^{2}f}{\partial y\partial x}$ are known as ‘mixed partial derivatives’. In general, mixed partial derivatives in the nth dimension commute if:

• They are taken with respect to independent variables in the same co-ordinate system

• The function f(x₁,x₂,…x_n) is continuous and differentiable in all its variables

In such a case:

$$\frac{\partial^{n}f}{\partial x_{1}\partial x_{2}\ldots\partial x_{n}} \equiv \ldots \equiv \frac{\partial^{n}f}{\partial x_{n}\partial x_{n - 1}\ldots\partial x_{1}}$$

Proof of commutivity for continuous, differentiable functions of two variables:

Let a=f(x,y),b=f(x+h,y),c=f(x,y+k),d=f(x+h,y+k):

$$\frac{\partial^{2}f}{\partial x\partial y} = \frac{\partial}{\partial x}\left( \frac{\partial f}{\partial y} \right) = \lim_{h \rightarrow 0}\frac{\lim_{k \rightarrow 0}\left( \frac{d - b}{k} \right) - \lim_{k \rightarrow 0}\left( \frac{c - a}{k} \right)}{h}$$

$$\frac{\partial^{2}f}{\partial y\partial x} = \frac{\partial}{\partial y}\left( \frac{\partial f}{\partial x} \right) = \lim_{k \rightarrow 0}\frac{\lim_{h \rightarrow 0}\left( \frac{d - c}{h} \right) - \lim_{h \rightarrow 0}\left( \frac{b - a}{h} \right)}{k}$$

So long as h,k0 is equivalent to k,h0 (the function f(x,y) is continuous with respect to both x and y):

$$\mathbf{\therefore}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{f}}{\mathbf{\partial x\partial y}}\mathbf{=}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{f}}{\mathbf{\partial y\partial x}}\mathbf{=}\underset{\mathbf{h,k \rightarrow 0}}{\mathbf{\lim}}{\frac{\mathbf{1}}{\mathbf{hk}}\mathbf{(d - c - b + a)}}$$

Integration of partial derivatives:

Integration of multivariable functions is carried out by the same principle as partial differentiation: the function is integrated with...

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