# Introduction To Partial Derivatives M, The Derivative Matrix

**Introduction To Partial Derivatives M, The Derivative Matrix**in

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## Functions of Several Variables

Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable.

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### Key Takeaways

Key PointsMultivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom.Unlike a single variable function **deterministic**: having exactly predictable time evolution**divergence**: a vector operator that measures the magnitude of a vector field’s source or sink at a given point, in terms of a signed scalar

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one. Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.

**A Scalar Field**: A scalar field shown as a function of

Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. Non-deterministic, or stochastic, systems can be studied using a different kind of mathematics, such as stochastic calculus. Quantitative analysts in finance also often use multivariate calculus to predict future trends in the stock market.

As we will see, multivariable functions may yield counter-intuitive results when applied to limits and continuity. Unlike a single variable function

We have also studied theorems linking derivatives and integrals of single variable functions. The theorems we learned are gradient theorem, Stokes’ theorem, divergence theorem, and Green’s theorem. In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes’ theorem, which applies to the integration of differential forms over manifolds.

## Limits and Continuity

A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.

### Learning Objectives

Describe the relationship between the multivariate continuity and the continuity in each argument

### Key Takeaways

Key PointsThe function **continuity**: lack of interruption or disconnection; the quality of being continuous in space or time**limit**: a value to which a sequence or function converges**scalar function**: any function whose domain is a vector space and whose value is its scalar field

A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single- variable functions. For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola. For example, the function

Continuity in each argument does not imply multivariate continuity. For instance, in the case of a real-valued function with two real-valued parameters,

It is easy to check that all real-valued functions (with one real-valued argument) that are given by

ight)

ight) = 1

## Partial Derivatives

A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.

### Learning Objectives

Identify proper ways to express the partial derivative

### Key Takeaways

Key PointsThe partial derivative of a function

ightarrow 0}{ f(a_1, dots, a_{i-1}, a_i+h, a_{i+1}, dots,a_n) – f(a_1, dots, a_i, dots,a_n) over h }**differential geometry**: the study of geometry using differential calculus**Euclidean**: adhering to the principles of traditional geometry, in which parallel lines are equidistant

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function *f* with respect to the variable *x* is variously denoted by

Suppose that *f* is a function of more than one variable. For instance, *one* of these lines and finding its slope*.* Usually, the lines of most interest are those which are parallel to the

To find the slope of the line tangent to the function at

So at

at the point

### Formal Definition

Like ordinary derivatives, the partial derivative is defined as a limit. Let

ightarrow R

ightarrow 0}{ f(a_1, cdots, a_{i-1}, a_i+h, a_{i+1}, cdots,a_n) – f(a_1, cdots, a_i, cdots,a_n) over h }}

## Tangent Planes and Linear Approximations

The tangent plane to a surface at a given point is the plane that “just touches” the surface at that point.

### Learning Objectives

Explain why the tangent plane can be used to approximate the surface near the point

### Key Takeaways

Key PointsFor a surface given by a differentiable multivariable function **differentiable**: having a derivative, said of a function whose domain and co-domain are manifolds**differential geometry**: the study of geometry using differential calculus**slope**: also called gradient; slope or gradient of a line describes its steepness

The tangent line (or simply the tangent) to a plane curve at a given point is the straight line that “just touches” the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that “just touches” the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized.

### Equations

When the curve is given by

where

The tangent plane to a surface at a given point

where

### Linear Approximation

Since a tangent plane is the best approximation of the surface near the point where the two meet, tangent plane can be used to approximate the surface near the point. The approximation works well as long as the point

## The Chain Rule

For a function

### Learning Objectives

Express a chain rule for a function with two variables

### Key Takeaways

Key PointsThe chain rule can be easily generalized to functions with more than two variables.For a single variable functions, the chain rule is a formula for computing the derivative of the composition of two or more functions. For example, the chain rule for **potential energy**: the energy possessed by an object because of its position (in a gravitational or electric field), or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)

The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if

The chain rule above is for single variable functions

This relation can be easily generalized for functions with more than two variables.

### Key Takeaways

Key PointsThe directional derivative is defined by the limit

abla_{mathbf{v}}{f}(mathbf{x}) = lim_{h

ightarrow 0}{frac{f(mathbf{x} + hmathbf{v}) – f(mathbf{x})}{h}}

abla_{mathbf{v}}{f}(mathbf{x}) =

abla f(mathbf{x}) cdot mathbf{v}**chain rule**: a formula for computing the derivative of the composition of two or more functions.**gradient**: of a function

The directional derivative of a multivariate differentiable function along a given vector

### Definition

The directional derivative of a scalar function

abla_{mathbf{v}}{f}(mathbf{x}) = lim_{h

ightarrow 0}{frac{f(mathbf{x} + hmathbf{v}) – f(mathbf{x})}{h}}}

If the function

abla_{mathbf{v}}{f}(mathbf{x}) =

abla f(mathbf{x}) cdot mathbf{v}

abla f(mathbf{x})

We can imagine the directional derivative

abla_{mathbf{v}}{f}(mathbf{x})

### Properties

Many of the familiar properties of the ordinary derivative hold for the directional derivative.

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### The Sum Rule

abla_mathbf{v} (f + g) =

abla_mathbf{v} f +

abla_mathbf{v} g

### The Constant Factor Rule

For any constant

abla_mathbf{v} (cf) = c

abla_mathbf{v} f

### The Product Rule (or Leibniz Rule)

abla_mathbf{v} (fg) = g

abla_mathbf{v} f + f

abla_mathbf{v} g

### The Chain Rule

If

abla_mathbf{v} hcirc g (p) = h”(g(p))

abla_mathbf{v} g (p)

## Maximum and Minimum Values

The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.

### Learning Objectives

Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

### Key Takeaways

Key PointsFor a function of two variables, the second partial derivative test is based on the sign of

ight)^2**critical point**: a maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined**intermediate value theorem**: a statement that claims that, for each value between the least upper bound and greatest lower bound of the image of a continuous function, there is a corresponding point in its domain that the function maps to that value**Rolle’s theorem**: a theorem stating that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero

The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).

### Finding Maxima and Minima of Multivariable Functions

The second partial derivative test is a method in multivariable calculus used to determine whether a critical point

For a function of two variables, suppose that

ight)^2

If

There are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function

## Lagrange Multiplers

The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.

### Key Takeaways

Key PointsTo maximize

abla_{x,y,lambda} Lambda(x, y, lambda)=0**gradient**: of a function **contour**: a line on a map or chart delineating those points which have the same altitude or other plotted quantity: a contour line or isopleth

In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.

For instance, consider the following optimization problem: Maximize

ight)

where the

ablaLambda = 0

### Introduction

One of the most common problems in calculus is that of finding maxima or minima (in general, “extrema”) of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside conditions or constraints. The method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables.

Consider the two-dimensional problem introduced above. Maximize

The contour lines of

and

abla_{x,y} f = – lambda

abla_{x,y} g

abla_{x,y} f= left( frac{partial f}{partial x}, frac{partial f}{partial y}

ight)

abla_{x,y} g= left( frac{partial g}{partial x}, frac{partial g}{partial y}

ight)

The constant is required because, although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. Note that

eq 0

To incorporate these conditions into one equation, we introduce an auxiliary function,

abla_{x,y,lambda} Lambda(x, y, lambda)=0

abla_{lambda} Lambda(x, y, lambda)=0

Where the Lagrange multiplier

Minimize

ight)

eq 0

eq 0

### Key Takeaways

Key PointsMathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives.An optimization process that involves only a single variable is rather straightforward. After finding out the function **optimization**: the design and operation of a system or process to make it as good as possible in some defined sense**cuboid**: a parallelepiped having six rectangular faces

Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. An optimization process that involves only a single variable is rather straightforward. After finding out the function

### Cardboard Box with a Fixed Volume

A packaging company needs cardboard boxes in rectangular cuboid shape with a given volume of 1000 cubic centimeters and would like to minimize the material cost for the boxes. What should be the dimensions

First of all, the material cost would be proportional to the surface area

We will first remove

ight)}

To find the critical points:

ight) = 0\ herefore y = frac{1000}{x^2}}

and

ight) = 0\ herefore x = frac{1000}{y^2}}

Then, substituting in the expression found equal to

Therefore, we find that:

That is to say, the box that minimizes the cost of materials while maintaining the desired volume should be a 10-by-10-by-10 cube.

## Applications of Minima and Maxima in Functions of Two Variables

Finding extrema can be a challenge with regard to multivariable functions, requiring careful calculation.

### Learning Objectives

Identify steps necessary to find the minimum and maximum in multivariable functions

### Key Takeaways

Key PointsThe second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.To find minima/maxima for functions with two variables, we must first find the first partial derivatives with respect to

ight)**multivariable**: concerning more than one variable**critical point**: a maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined

We have learned how to find the minimum and maximum in multivariable functions. As previously mentioned, finding extrema can be a challenge with regard to multivariable functions. In particular, we learned about the second derivative test, which is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum, using the value of the second derivative at the point. In this atom, we will find extrema for a function with two variables.

### Example

Find and label the critical points of the following function:

Plot of

To solve this problem we must first find the first partial derivatives of the function with respect to

ight)}

Looking at

We plug the first solution,

ight)\ ,quad = x^2}

There were other possibilities for

ight) \ ,quad= x(1-x)\ ,quad= 0}

So

ight) \ ,quad= x^2(8x-3) \ ,quad= 0}

So

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Let’s list all the critical values now:

ight)}}

Now we have to label the critical values using the second derivative test. Plugging in all the different critical values we found to label them, we have:

ight) = 0.210938

We can now label some of the points:

at (0, −1),

ight)

At the remaining point we need higher-order tests to find out what exactly the function is doing.

**Derivative**