04 Partial Derivatives

Functions of Several Variables¶

Let \(D\) be the set of all \(n\) tuples of the form \((x_1, x_2, \dots , x_n)\), where \(x_1, x_2, \dots, x_n\) are real numbers. A function on \(D\) is a rule \(f\) that assigns a number \(w = f(x_1, x_2, \dots, x_n)\) for each element in \(D\).

If there exists only number \(w\) for each element in \(D\), then it is said to be a single-valued function. If more than one \(w\) exists, then it is said to be a many-valued function.

While finding domain \(D\)

we include all the points which make the function \(f\) well-defined
neglect the values which make \(w\) a complex or undefined number

Neighborhood¶

A neighborhood of a point \(P_0(x_0, y_0)\) is a circular disc, with centre @ \(P_0\) and radius \(r\), where \(r\) is a small +ve number.

If

\(r= \epsilon\), \(\epsilon\) neighborhood
\(r = \delta\), \(\delta\) neighborhood

In 3 dimensions, we replace circular disk with an open spherical ball with centre @ \(P_0\)

Types of points¶

Let \(S\) be a non-empty set in the XY plane . A point \(P_0(x_0, y_0)\) is said to be

Point	Condition
Interior	there exists a neighborhood of \(P_0\) which lies completely inside \(S\)
Boundary	every neighborhood of \(P_0\) contains points of \(S\) and points outside \(S\)
Exterior	there exists a neighborhood of \(P_0\) completely outside \(S\)

Types of Sets¶

	Characteristic
Open	contains interior points only
Closed	contains interior and all boundary points
Bounded	lies completely inside an open disk of finite radius
Unbounded	cannot be enclosed inside open disk of finite radius

\(XY\) plane is both open and closed.

Level¶

For a function \(f(x, y)\) and constant \(c\),

	Equation
Level Curve	\(f(x, y) = c\)
Level Surface	\(f(x, y, z) = c\)

Limits¶

Let \(f\) be a function defined at all points in the some neighborhood f \((x_0, y_0)\). We say that \(f\) has a limit \(L\), when the point \((x, y)\) approaches \((x_0, y_0)\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that

\[ \begin{aligned} 0 < \text{ D b/w } (x, y) \text{ and } (x_0, y_0) &< \delta \\ 0 < \sqrt{ (x-x_0)^2 + (y-y_0)^2 } &< \delta \\ | f(x,y) - L | &< \epsilon \\ \implies L &= \lim_{(x, y) \to (x_0, y_0)} f(x, y) \end{aligned} \]

Here, \((x, y)\) approaches \((x_0, y_0)\) in an infinite number of ways.

2 Path Test¶

TO show that the limit of \(f(x, y)\) does not exist @ \((x_0, y_0)\), we find 2 different paths through which the value of limits are different.

We choose the path as \(y = mx^n\) or \(x = m y^n\), where \(m\) and \(n\) are constants. The choice depends on the problem. We try to obtain a final limit in terms of \(m\)

Continuity¶

A function \(f(x, y)\) is continuous at \((x_0, y_0)\) if

\(f(x_0, y_0)\) exists
\(\lim_{(x, y) \to (x_0, y_0)} f(x,y)\) exists
\(\lim_{(x, y) \to (x_0, y_0)} f(x,y) = f(x_0, y_0)\)

The following functions are continuous in their domain of definition

Polynomial
Exponential
Circular
Trignometric

Partial Derivatives¶

Let \(f(x,y)\) be a function of 2 variables.

Provided the limit exists, the partial derivative of \(f\) wrt \(x\) is denoted and defined by

\[ \begin{aligned} \frac{\partial f}{\partial x} &= \lim_{\Delta x \to 0} \frac{ f(x + \Delta x, \ y) - f(x, y) }{\Delta x} \\ \frac{\partial f}{\partial y} &= \lim_{\Delta y \to 0} \frac{ f(x, \ y + \Delta y) - f(x, y) }{\Delta y} \end{aligned} \]

We define higher order partial derivatives as

\[ \begin{aligned} f_x &= \frac{\partial^2 f}{\partial x^2} &= \frac{\partial}{\partial x}\left[ \frac{\partial f}{\partial x} \right] \\ f_{xy} &=\frac{\partial^2 f}{\partial x \partial y} &= \frac{\partial}{\partial x}\left[ \frac{\partial f}{\partial y} \right] \\ f_{xy} &= f_{yx} \\ f_{xx} &= (f_x)_x \end{aligned} \]

Laplace Equation¶

If \(u\) is a function

\[ u_{xx} + u_{yy} + u_{zz} = 0 \]

Chain Rule¶

If \(w = f(x, y)\) a function where \(x, y\) are themselves functions of

an independent parameter \(t\)

\[ \frac{dw}{dt} = \left( \frac{\partial w}{\partial x} \cdot \frac{dx}{dt} \right) + \left( \frac{\partial w}{\partial y} \cdot \frac{dy}{dt} \right) \]

2 independent parameters \(u, v\)

\[ \begin{aligned} \frac{\partial w}{\partial u} &= \left( \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial u} \right) + \left( \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial u} \right) \\ \frac{\partial w}{\partial v} &= \left( \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial v} \right) + \left( \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial v} \right) \end{aligned} \]

Implicit Differentiation¶

Let \(y\) be a function of \(x\), expressed as an implicit relation \(f(x, y) = 0\).

Differentiating partially wrt \(x\)

\[ \begin{aligned} \frac{\partial f}{\partial x} + \left( \frac{\partial f}{\partial y} \cdot \frac{dy}{dx} \right) &= 0 \\ \implies \frac{dy}{dx} &= \frac{-\partial f / \partial x}{\partial f / \partial y} \\ &= \frac{- f_x}{f_y} \end{aligned} \]

If \(z\) is a function of \(x\) and \(y\), given by an implicit relation \(f(x,y,z) = 0\)

\[ \begin{aligned} z_x &= \frac{-f_x}{f_z} \\ z_y &= \frac{-f_y}{f_z} \end{aligned} \]

Gradient Vector¶

Let \(f = f(x,y)\) be a function. Then the gradient of \(f\)

\[ \begin{aligned} \text{grad } f &= \nabla f \\ &= f_x \cdot \hat i + f_y \cdot \hat j \end{aligned} \]

\(\nabla\) is the vector differential operator

\(\nabla f\) acts along the normal at any point to the level curve of \(f\)

Directional Derivative¶

Let \(f\) be a function defined at all pionts in some neighborhood of \(P_0(x_0, y_0)\). Then, provided the limit exists, the directional derivative of \(f\) in the direction of \(\vec a = a_1 \hat i + a_2 \hat j\) is given by

\[ \begin{aligned} \text{DD} &= (D_{\hat u} f)_{P_0} \\ &= \lim_{s \to 0} \frac{f(x_0 + su_1, y_0 + su_2) - f(x_0, y_0)}{s} \\ &= \nabla f \cdot \hat u \\ \nabla f &= ( \nabla f )_{P_0} \\ \hat u &= u_1 \hat i + u_2 \hat j, \text{ unit vector in direction of } \vec a \\ &= \frac{\vec A}{|\vec A|} \end{aligned} \]

Notes¶

Direction	f	DD
\(\nabla f\)	increases more rapidly	\(\vert \nabla f \vert\)
\(- \nabla f\)	decreases more rapidly	\(- \vert \nabla f \vert\)
\(\perp \text{to } (\nabla f) \text{ or } (-\nabla f)\)	no change	0

Tangent Plane¶

Let \(f = f(x, y, z)\). Then, the equation of the tangent plane passing through a point \(P_0(x_0, y_0, z_0)\) is given by

\[ (x - x_0) {f_x}_{(P_0)} + (y - y_0) {f_y}_{(P_0)} + (z - z_0) {f_z}_{(P_0)} = 0 \]

Normal Line¶

The equations of normal line at \(P_0\) are given by

\[ \begin{aligned} x &= x_0 + t {f_x}_{(P_0)} \\ y &= y_0 + t {f_y}_{(P_0)} \\ z &= z_0 + t {f_z}_{(P_0)} \end{aligned}, \quad t \text{ is some parameter} \]

Linearisation¶

Let \(f(x, y, z)\) be a function and \(P_0(x_0, y_0, z_0)\) be any point in the domain of definition. Then, the linearisation of \(f\) about \(P_0\) is given by

\[ L(x, y, z) = f(P_0) + (x - x_0){f_x}_{(P_0)} + (y - y_0){f_y}_{(P_0)} + (z - z_0){f_z}_{(P_0)} \]

At all continuous points, \(f\) and \(L\) are the same.

Extreme Values of a Function¶

Let \(f(x,y)\) be a function, and \((a,b)\) be a point.

Absolute maximum is the point at which \(f\) is max; absolute minimum is the point at which \(f\) is minimum. They are obtained by evaluating \(f\) at all local minima/maxima and comparing the values.

Local Point	Characteristic
Maximum	\(f(a, b) > f(x, y), \quad \forall (x, y)\) in the neighborhood of \((a,b)\)
Minimum	\(f(a, b) < f(x, y), \quad \forall (x, y)\) in the neighborhood of \((a,b)\)
Saddle	\(f\) increases in some directions and decreases in other directions at \((a, b)\)

Finding local points¶

At point \((a, b)\)

\[ \begin{aligned} 1. & f_x = 0 \text{ and } f_y = 0 \\ 2. & r = f_{xx}, s = f_{xy}, t = f_{yy}, \\ & D = rt - s^2 \end{aligned} \]

\(D\)	\(r\)	\((a, b)\)
> 0	< 0	Maximum
> 0	> 0	Minimum
< 0	-	Saddle
= 0	-	Test Fails

Note: In the above table, we can replace \(r\) by \(t\) as well.

Constrained maxima, minima¶

We extremise a function \(f(x, y, z)\) subject to constraint/condition \(\phi(x, y, z) = 0\). We then proceed as follows

From Lagrange’s function, \(\lambda =\) Lagrange’s multiplier constant

\[ F(x, y, z) = f + \lambda \phi \]

The extreme values are given by

\[ F_x = F_y = F_z = 0 \]

Solve the equations for \(x, y, z, \lambda\)

Note

By Lagrange’s method, we cannot find whether \(f\) has a maximum or minimum
If \(f\) is to be extremised subject to constraints \(\phi_1 = \phi_2 = 0\), then the Lagrange’s function becomes

\[ F = f + \lambda_1 \phi_1 + \lambda_2 \phi_2 \]

Last Updated: 2024-01-24 ; Contributors: AhmedThahir