Contents
1 Functions Of Several Variables
1.1 Level Curves
1.2 Limits and Continuity
1.3 Directional Derivatives and Gradient Vector
1.4 Maximising the Directional Derivative
1.5 Maximum and Minimum Values
1.6 Extrema for Functions with Side Conditions (Constraints)
1.7 Absolute Maximum and Minimum Values
1.8 Taylor’s Formula For Functions of Several Variables
1.9 Composite Functions and Chain Rule Differentiation
1.10 Euler’s Theorem on Homogeneous Functions
1.11 Implicit Functions and Jacobians
1.12 Jacobians
1.13 Implicit Differentiation
1.14 Implicit Function Theorem
1.15 Functions From \(\mathbb {R}^n\) To \(\mathbb {R}^m\)
1.16 Inverse Function Theorem
1.17 Functional Dependence
1.18 Practice Problems
2 Surfaces
2.1 Surface of Revolution
2.2 Cylindrical and Spherical Coordinates
2.3 Tangent Planes
2.4 Parametric Surfaces
2.5 Tangent Planes To Parametric Surfaces
2.6 Tangent Planes To Level Curves
2.7 Practice Problems
3 Multiple Integration
3.1 Volumes and Double Integration
3.2 Iterated Integrals
3.3 Double Integrals Over General Regions
3.4 Properties Of Double Integrals
3.5 Double Integrals In Polar Coordinates
3.6 Applications Of Double Integrals
3.7 Triple Integrals
3.8 Fubini’s Theorem for Triple Integrals
3.9 Triple Integrals Over A General Bounded Region \(E\)
3.10 Spherical Coordinates
3.11 Applications Of Triple Integrals
3.12 Triple Integrals In Cylindrical Coordinates
3.13 Change of Variables
3.14 Surface Area
3.15 Practice Problems
4 Vector Analysis
4.1 Vector Fields
4.2 Gradients
4.3 The Operator \(\nabla \)
4.4 divergence and curl of a Vector Field
4.5 Line Integrals
4.6 Line Integrals Of Vector Fields
4.7 The Fundamental Theorem of Line Integrals
4.8 Transformation Of Line Integrals Into Double Integrals
4.9 Parametric Surfaces and their Areas
4.10 Surface Integrals
4.11 Surface Integrals of Vector Fields
4.12 Oriented Surfaces
4.13 Stokes’ Theorem
4.14 The divergence Theorem
4.15 Practice Problems
5 Differential Equations
5.1 Systems of Differential Equations
5.2 Solution by Elimination
5.3 Undetermined Coefficients
5.4 Laplace Transformation
5.5 Using Differential Operators
5.6 The Power Series Method
5.7 The Method Of Frobenius
5.8 Practice Problems
6 Laplace Transforms
6.1 Linearity
6.2 Laplace Transform of the Derivatives of \(f(t)\)
6.3 Laplace Transform of the Integral of a Function
6.4 First Shifting Theorem: the \(s\)-Shift
6.5 Second Shifting Theorem: the \(t\)-Shift
6.6 Differentiation of Transforms
6.7 Convolution Theorem
6.8 Integral Equations
6.9 Integration of Transforms
6.10 Partial Fractions
6.11 Practice Problems
7 Fourier Series and Integrals
7.1 Periodic Functions and Trigonometric Series
7.2 Orthogonality of the Trigonometric System
7.3 Fourier Series
7.4 Convergence and Sum of Fourier Series
7.5 Functions of Any Period \(P=2L\)
7.6 Even and Odd Functions
7.7 Practice Problems
References
8 Revision Problems
8.1 From Test 1
8.2 From Test 2
8.3 From Test 3
1.1 Level Curves
1.2 Limits and Continuity
1.3 Directional Derivatives and Gradient Vector
1.4 Maximising the Directional Derivative
1.5 Maximum and Minimum Values
1.6 Extrema for Functions with Side Conditions (Constraints)
1.7 Absolute Maximum and Minimum Values
1.8 Taylor’s Formula For Functions of Several Variables
1.9 Composite Functions and Chain Rule Differentiation
1.10 Euler’s Theorem on Homogeneous Functions
1.11 Implicit Functions and Jacobians
1.12 Jacobians
1.13 Implicit Differentiation
1.14 Implicit Function Theorem
1.15 Functions From \(\mathbb {R}^n\) To \(\mathbb {R}^m\)
1.16 Inverse Function Theorem
1.17 Functional Dependence
1.18 Practice Problems
2 Surfaces
2.1 Surface of Revolution
2.2 Cylindrical and Spherical Coordinates
2.3 Tangent Planes
2.4 Parametric Surfaces
2.5 Tangent Planes To Parametric Surfaces
2.6 Tangent Planes To Level Curves
2.7 Practice Problems
3 Multiple Integration
3.1 Volumes and Double Integration
3.2 Iterated Integrals
3.3 Double Integrals Over General Regions
3.4 Properties Of Double Integrals
3.5 Double Integrals In Polar Coordinates
3.6 Applications Of Double Integrals
3.7 Triple Integrals
3.8 Fubini’s Theorem for Triple Integrals
3.9 Triple Integrals Over A General Bounded Region \(E\)
3.10 Spherical Coordinates
3.11 Applications Of Triple Integrals
3.12 Triple Integrals In Cylindrical Coordinates
3.13 Change of Variables
3.14 Surface Area
3.15 Practice Problems
4 Vector Analysis
4.1 Vector Fields
4.2 Gradients
4.3 The Operator \(\nabla \)
4.4 divergence and curl of a Vector Field
4.5 Line Integrals
4.6 Line Integrals Of Vector Fields
4.7 The Fundamental Theorem of Line Integrals
4.8 Transformation Of Line Integrals Into Double Integrals
4.9 Parametric Surfaces and their Areas
4.10 Surface Integrals
4.11 Surface Integrals of Vector Fields
4.12 Oriented Surfaces
4.13 Stokes’ Theorem
4.14 The divergence Theorem
4.15 Practice Problems
5 Differential Equations
5.1 Systems of Differential Equations
5.2 Solution by Elimination
5.3 Undetermined Coefficients
5.4 Laplace Transformation
5.5 Using Differential Operators
5.6 The Power Series Method
5.7 The Method Of Frobenius
5.8 Practice Problems
6 Laplace Transforms
6.1 Linearity
6.2 Laplace Transform of the Derivatives of \(f(t)\)
6.3 Laplace Transform of the Integral of a Function
6.4 First Shifting Theorem: the \(s\)-Shift
6.5 Second Shifting Theorem: the \(t\)-Shift
6.6 Differentiation of Transforms
6.7 Convolution Theorem
6.8 Integral Equations
6.9 Integration of Transforms
6.10 Partial Fractions
6.11 Practice Problems
7 Fourier Series and Integrals
7.1 Periodic Functions and Trigonometric Series
7.2 Orthogonality of the Trigonometric System
7.3 Fourier Series
7.4 Convergence and Sum of Fourier Series
7.5 Functions of Any Period \(P=2L\)
7.6 Even and Odd Functions
7.7 Practice Problems
References
8 Revision Problems
8.1 From Test 1
8.2 From Test 2
8.3 From Test 3