Engineering Math

11.3.1 The Directional Derivative

The directional derivative is just what its name suggests. It is the derivative of a function in a particular direction. The following picture illustrates the situation in the case of a function of two variables.

In this picture, v ≡

is a unit vector in the xy plane and x0 ≡
is a point in the xy plane. When
moves in the direction of v, this results in a change in z = f
as shown in the picture. The directional derivative in this direction is defined as

It tells how fast z is changing in this direction. If you looked at it from the side, you would be getting the slope of the indicated tangent line. A simple example of this is a person climbing a mountain. He could go various directions, some steeper than others. The directional derivative is just a measure of the steepness in a given direction. This motivates the following general definition of the directional derivative.

Definition 11.3.1 Let f : U → ℝ where U is an open set in ℝn and let v be a unit vector. For x ∈ U, define the directional derivative of f in the direction v, at the point x as

Example 11.3.2 Find the directional derivative of the function f

= x2y in the direction of i + j at the point
.

First you need a unit vector which has the same direction as the given vector. This unit vector is v ≡

. Then to find the directional derivative from the definition, write the difference quotient described above. Thus f
=
2
and f
= 2. Therefore,

and to find the directional derivative, you take the limit of this as t → 0. However, this difference quotient equals

and so, letting t → 0,Dvf
=
.

There is something you must keep in mind about this. The direction vector must always be a unit vector1 .

Chapter August 25, 2018 1 Some Prerequisite Topics

1.1 Sets And Set Notation
1.2 Well Ordering And Induction
1.3 The Complex Numbers
1.4 Polar Form Of Complex Numbers
1.5 Roots Of Complex Numbers
1.6 The Quadratic Formula
1.7 The Complex Exponential
1.8 Dividing Polynomials
1.9 The Fundamental Theorem Of Algebra
1.10 Exercises

Part I  Linear Algebra And Multivariable Calculus

Chapter 2 Fundamentals

2.1 Fn
2.2 Algebra in ℝn
2.3 Geometric Meaning Of Vector Addition In ℝ3
2.4 Lines
2.5 Distance in ℝn
2.6 Geometric Meaning Of Scalar Multiplication In ℝ3
2.7 Exercises
2.8 Physical Vectors
2.9 Exercises

Chapter 3 Vector Products

3.1 The Dot Product
3.2 The Geometric Significance Of The Dot Product
3.2.1 The Angle Between Two Vectors
3.2.2 Work And Projections
3.2.3 The Dot Product And Distance In ℂn
3.3 Exercises
3.4 The Cross Product
3.4.1 The Box Product
3.5 Proof of the distributive law
3.5.1 Torque
3.5.2 Center Of Mass
3.5.3 Angular Velocity
3.6 Vector Identities And Notation
3.7 Planes
3.8 Exercises

Chapter 4 Systems Of Equations

4.1 Systems Of Equations, Algebraic Procedures
4.1.1 Elementary Operations
4.1.2 Gauss Elimination
4.1.3 Balancing Chemical Reactions
4.1.4 Dimensionless Variables∗
4.2 MATLAB And Row Reduced Echelon Form
4.3 Exercises

Chapter 5 Matrices

5.1 Linear Transformations and Matrices
5.2 Multiplication of Matrices
5.2.1 The Transpose
5.3 Some Examples of Linear Functions on ℝn
5.3.1 Rotations in ℝ2
5.3.2 Projections
5.3.3 Rotations About A Particular Vector
5.4 The Inverse of a Matrix
5.4.1 The Identity And Inverses
5.4.2 Finding The Inverse Of A Matrix
5.5 MATLAB And Matrix Arithmetic
5.6 Exercises

Chapter 6 Subspaces Spans and Bases

6.1 Subspaces
6.2 Exercises

Chapter 7 Matrices and The Inner Product

7.1 Eigenvalues and Eigenvectors
7.2 Using Matlab
7.3 Distance and Unitary Matrices
7.4 Schur’s Theorem
7.5 Diagonalization
7.6 Approximations
7.6.1 Fredholm Alternative
7.6.2 Least Squares
7.6.3 Regression lines
7.6.4 Identifying the Closest Point
7.6.5 Using Matlab
7.7 The Singular Value Decomposition∗
7.8 Exercises

Chapter 8 Vector Valued Functions

8.1 Vector Valued Functions
8.2 Vector Fields
8.3 Exercises
8.4 Continuous Functions
8.4.1 Sufficient Conditions For Continuity
8.5 Limits Of A Function
8.6 Properties Of Continuous Functions
8.7 Exercises
8.8 Open And Closed Sets
8.9 Exercises
8.10 Some Fundamentals∗
8.10.1 Combinations Of Continuous Functions
8.10.2 The Nested Interval Lemma
8.10.3 Convergent Sequences, Sequential Compactness
8.10.4 Continuity And The Limit Of A Sequence
8.10.5 The Extreme Value Theorem And Uniform Continuity
8.10.6 Convergence of Functions
8.11 Root Test
8.12 Convergence of Sums
8.13 Exercises

Chapter 9 Vector Valued Functions Of One Variable

9.1 Limits Of A Vector Valued Function Of One Real Variable
9.2 The Derivative And Integral
9.2.1 Geometric And Physical Significance Of The Derivative
9.2.2 Differentiation Rules
9.2.3 Leibniz’s Notation
9.3 Exercises
9.4 Line Integrals
9.4.1 Arc Length And Orientations
9.4.2 Line Integrals And Work
9.4.3 Another Notation For Line Integrals
9.5 Exercises
9.6 Independence Of Parametrization∗
9.6.1 Hard Calculus
9.6.2 Independence Of Parametrization

Chapter 10 Motion On A Space Curve

10.1 Space Curves
10.1.1 Some Simple Techniques
10.2 Geometry Of Space Curves∗
10.3 Exercises

Chapter 11 Functions Of Many Variables

11.1 Review Of Limits
11.2 Exercises
11.3 The Directional Derivative And Partial Derivatives
11.3.1 The Directional Derivative
11.3.2 Partial Derivatives
11.4 Exercises
11.5 Mixed Partial Derivatives
11.6 Partial Differential Equations
11.7 Exercises

Chapter 12 The Derivative Of A Function Of Many Variables

12.1 The Derivative Of Functions Of One Variable, o(v)
12.2 Exercises
12.3 The Derivative Of Functions Of Many Variables
12.4 Exercises
12.5 C1 Functions
12.6 The Chain Rule
12.6.1 The Chain Rule For Functions Of One Variable
12.6.2 The Chain Rule For Functions Of Many Variables
12.7 Exercises
12.7.1 Related Rates Problems
12.7.2 The Derivative Of The Inverse Function
12.7.3 Proof Of The Chain Rule
12.8 Exercises
12.9 The Gradient
12.10 The Gradient And Tangent Planes
12.11 Exercises

Chapter 13 Optimization

13.1 Local Extrema
13.2 Exercises
13.3 The Second Derivative Test
13.4 Exercises
13.5 Lagrange Multipliers
13.6 Exercises
13.7 Proof Of The Second Derivative Test∗

Chapter 14 The Riemannn Integral On ℝn

14.1 Methods For Double Integrals
14.1.1 Density And Mass
14.2 Exercises
14.3 Methods For Triple Integrals
14.3.1 Definition Of The Integral
14.3.2 Iterated Integrals
14.4 Exercises
14.4.1 Mass And Density
14.5 Exercises

Chapter 15 The Integral In Other Coordinates

15.1 Polar Coordinates
15.2 Exercises
15.3 Cylindrical And Spherical Coordinates
15.3.1 Volume and Integrals in Cylindrical Coordinates
15.3.2 Volume And Integrals in Spherical Coordinates
15.4 Exercises
15.5 The General Procedure
15.6 Exercises
15.7 The Moment Of Inertia And Center Of Mass
15.8 Exercises

Chapter 16 The Integral On Two Dimensional Surfaces In ℝ3

16.1 The Two Dimensional Area In ℝ3
16.1.1 Surfaces Of The Form z = f (x,y)
16.2 Flux Integrals
16.3 Exercises

Chapter 17 Calculus Of Vector Fields

17.1 Divergence And Curl Of A Vector Field
17.1.1 Vector Identities
17.1.2 Vector Potentials
17.1.3 The Weak Maximum Principle
17.2 Exercises
17.3 The Divergence Theorem
17.3.1 Coordinate Free Concept Of Divergence
17.4 Some Applications Of The Divergence Theorem
17.4.1 Hydrostatic Pressure
17.4.2 Archimedes Law Of Buoyancy
17.4.3 Equations Of Heat And Diffusion
17.4.4 Balance Of Mass
17.4.5 Balance Of Momentum
17.4.6 Frame Indifference
17.4.7 Bernoulli’s Principle
17.4.8 The Wave Equation
17.4.9 A Negative Observation
17.4.10 Volumes Of Balls In ℝn
17.4.11 Electrostatics
17.5 Exercises

Chapter 18 Stokes And Green’s Theorems

18.1 Green’s Theorem
18.2 Exercises
18.3 Stoke’s Theorem From Green’s Theorem
18.3.1 The Normal And The Orientation
18.3.2 The Mobeus Band
18.4 A General Green’s Theorem
18.4.1 Conservative Vector Fields
18.4.2 Some Terminology

Chapter 19 Implicit Function Theorem*

19.1 More Continuous Partial Derivatives
19.2 The Method Of Lagrange Multipliers
19.3 The Local Structure Of C1 Mappings

Chapter 20 The Theory Of The Riemannn Integral∗

20.1 An Important Warning
20.2 Basic Definition
20.3 Basic Properties
20.4 Which Functions Are Integrable?
20.5 Iterated Integrals
20.6 The Change Of Variables Formula
20.7 Some Observations

Part II  Linear Algebra and Differential Equations

Chapter 21 Determinants

21.1 Basic Techniques And Properties
21.1.1 Cofactors And 2 × 2 Determinants
21.1.2 The Determinant Of A Triangular Matrix
21.1.3 Properties Of Determinants
21.1.4 Finding Determinants Using Row Operations
21.2 Applications
21.2.1 A Formula For The Inverse
21.2.2 Finding Eigenvalues Using Determinants
21.2.3 Cramer’s Rule
21.3 MATLAB And Determinants
21.4 The Cayley Hamilton Theorem∗
21.5 Exercises

Chapter 22 The Mathematical Theory Of Determinants∗

22.0.1 The Function sgn
22.1 The Determinant
22.1.1 The Definition
22.1.2 Permuting Rows Or Columns
22.1.3 A Symmetric Definition
22.1.4 The Alternating Property Of The Determinant
22.1.5 Linear Combinations And Determinants
22.1.6 The Determinant Of A Product
22.1.7 Cofactor Expansions
22.1.8 Formula For The Inverse
22.1.9 Cramer’s Rule

Chapter 23 Methods and Recipes for First Order ODE

23.1 First Order Linear Equations
23.2 Bernouli Equations
23.3 Separable Differential Equations, Stability
23.4 Homogeneous Equations
23.5 Exact Equations
23.6 The Integrating Factor
23.7 The Case Where M,N Are Affine Linear
23.8 Linear and Nonlinear Differential Equations
23.9 Computer Algebra Methods
23.9.1 Maple
23.9.2 Mathematica
23.9.3 MATLAB
23.9.4 Scientific Notebook
23.10 Exercises

Chapter 24 First Order Linear Systems

24.1 The Theory of the Fundamental Matrix
24.2 Laplace Transform Methods
24.3 Using a Computer Algebra System
24.4 Using Computer Algebra to Find Fundamental Matrix
24.5 Homogeneous Particular and General Solutions
A.1 Gamma Function
A.2 Laplace Transforms
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