Calculus of Scalar and Vector Fields

Do you want to learn how to manipulate and analyze scalar and vector fields using advanced calculus techniques? Do you want to understand the concepts and methods of scalar and vector products, gradient, divergence, curl, and Laplacian operators? Do you want to apply the principles of vector analysis to solve problems in physics, engineering, and other disciplines? If you answered yes to any of these questions, then this course is for you! In this course, you will learn the fundamentals of calculus of scalar and vector fields, which is the branch of mathematics that deals with functions and operators that map points in space to scalars or vectors. You will learn how to use different notations and conventions to express scalar and vector fields, such as Einstein's summation, Kronecker delta, and Levi-Civita symbols. You will learn how to perform scalar and vector products of vectors, which are useful for measuring angles, areas, and volumes. You will learn how to compute and interpret the gradient of a scalar field, which measures the rate and direction of change of a scalar function. You will learn how to compute and interpret the divergence and curl of a vector field, which measure the sources and vortices of a vector field. You will also learn how to use the Laplacian operator, which combines the divergence and the gradient, and is useful for studying heat conduction, electrostatics, and fluid dynamics. This course will equip you with the skills and knowledge to solve various problems involving scalar and vector fields. You will be able to apply the concepts and methods of calculus of scalar and vector fields to fields such as physics, engineering, chemistry, biology, and more. You will also be able to appreciate the power of vector analysis theorems, such as Gauss', Green's, and Stokes' theorems, which relate the integrals of scalar and vector fields over different domains. By the end of this course, you will be able to: - Define and explain the meaning of scalars, vectors, and tensors - Identify and use different notations and conventions for scalar and vector fields - Perform scalar and vector products of vectors and calculate angles, areas, and volumes - Compute and interpret the gradient of a scalar field and its properties - Compute and interpret the divergence and curl of a vector field and their properties - Compute and interpret the Laplacian of a scalar or a vector field and its applications - Use different coordinate systems and bases to express scalar and vector fields and operators - Compute and interpret the gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates - Recognize and use some special curvilinear coordinates, such as cylindrical and spherical coordinates - Apply the vector analysis theorems, such as Gauss', Green's, and Stokes' theorems, to relate the integrals of scalar and vector fields over different domains Created for students, engineers, scientists, and anyone passionate about learning the calculus of scalar and vector fields, this course assumes a basic background in calculus, linear algebra, and vector algebra. With engaging video lessons, interactive quizzes, practice problems, and supportive peer interactions, you'll embark on a journey of discovery and mastery like never before. Seize the opportunity to advance your mathematical skills and unlock new problem-solving possibilities with an enrolment right away!

133

36 hrs

$ 10.00

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

MTH 202: Mathematical Methods II

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

CHE 305: Engineering Analysis I

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables. Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

Introduction

Meaning of scalars, vectors and tensors; Einstein's summation convention, Kronecker delta and Levi-Civita notations; orthogonal and orthonormal vector bases.

Chapter lessons

1.Scalars, vectors and tensors29:01

Meaning and examples of scalars, tensors and vectors.

2.Einstein's summation52:23

Einstein's summation convention and its use.

3.Kronecker delta24:01

Kronecker delta notation and its use.

4.Levi-Civita10:40

Levi-Civita notation and its use.

5.Orthogonal and orthonormal bases29:05

Meaning of orthogonal and orthonormal bases of a vector space.

Scalar and Vector Products

Scalar (dot) and vector (cross) products of vectors using sign functions.

Chapter lessons

1.Scalar products (1)17:50

Scalar product of two vectors - using their magnitudes and angle, and using their Cartesian components.

2.Scalar products (2)15:24

Scalar product of two vectors - using sign notations.

3.Vector products (1)30:58

Vector product of two vectors - using their magnitudes and angle, and using their Cartesian components.

4.Vector products (2)34:04

Vector product of two vectors - using sign notations.

5.Worked examples (1)21:45

Worked examples on scalar and vector products using sign functions.

6.Worked examples (2)20:49

More worked examples on scalar and vector products using sign functions.

Scalar and Vector Fields

Definitions, examples and some visualizations of scalar and vector fields.

Chapter lessons

1.Scalar fields12:10

Meaning and examples of scalar fields.

2.Vector fields14:49

Meaning and examples of vector fields.

3.Visualization33:43

Visualizing some plane vector fields.

Gradient of a Scalar Field

Definition, evaluation and properties of the gradient of a scalar function.

Chapter lessons

1.Definition21:40

Meaning of the gradient of a scalar field, potential or gradient fields.

2.Illustration38:33

Making sense of the gradient of a scalar - its implications and how it relates to the derivative of single-variable functions.

3.Directional derivative43:55

Meaning of directional derivatives, their maximum values, and how they relate to gradients.

4.Level curves and surfaces23:14

Meaning of level curves and surfaces, and their relation to the gradient.

5.Properties10:09

Properties of the gradient.

6.Worked examples (1)10:12

Worked examples on gradients, directional derivatives and normals to surfaces.

7.Worked examples (2)28:07

More worked examples on gradients, directional derivatives and normals to surfaces.

8.Worked examples (3)50:33

More worked examples on gradients, directional derivatives and normals to surfaces.

9.Worked examples (4)14:19

More worked examples on gradients, directional derivatives and normals to surfaces.

Divergence of a Vector Field

Definition, evaluation and properties of the divergence of a vector field, and an introduction to the Laplacian of scalar and vector fields.

Chapter lessons

1.Definition10:12

Meaning of the divergence of a vector field and solenoidality.

2.Illustration13:37

Making sense of the divergence of a vector field.

3.Laplacian9:17

Meaning of the Laplacian of scalar and vector fields.

4.Properties19:19

Properties of the divergence of a vector field.

5.Worked examples (1)15:58

Worked examples on the divergence of vector fields and some Laplacian.

6.Worked examples (2)18:31

More worked examples on the divergence of vector fields and some Laplacian.

7.Worked examples (3)45:13

More worked examples on the divergence of vector fields and some Laplacian.

Curl of a Vector Field

Definition, evaluation and properties of the curl (spin or rotor) of a vector field.

Chapter lessons

1.Definition10:12

Meaning of the curl of a vector field and irrotationality.

2.Illustration7:45

Making sense of the curl of a vector field.

3.Properties28:47

Properties of the curl of a vector field.

4.Worked examples (1)33:05

Worked examples on the curl of a vector field.

5.Worked examples (2)30:42

More worked examples on the curl of a vector field.

6.Worked examples (3)35:23

More worked examples on the curl of a vector field.

Use of Sign Functions

Definitions and operations of gradient of a scalar field and divergence and curl of a vector field using sign functions and conventions such as the Einstein summation, Kronecker delta and Levi-Civita notations.

Chapter lessons

1.Definitions33:16

Defining the gradient, divergence, curl and Laplacian using sign conventions.

2.Worked examples (1)30:27

Worked examples on the use of sign conventions for gradient, divergence, curl and Laplacian.

3.Worked examples (2)15:11

Worked examples on the use of sign conventions for gradient, divergence, curl and Laplacian.

4.Worked examples (3)34:47

Worked examples on the use of sign conventions for gradient, divergence, curl and Laplacian.

Orthogonal Curvilinear Coordinates (1)

Definitions and illustrations of coordinates, coordinate lines and coordinate surfaces for the Cartesian coordinate system, general curvilinear coordinates, and general orthogonal curvilinear coordinates.

Chapter lessons

1.Cartesian coordinates43:50

Coordinate points, lines and surfaces on the Cartesian coordinate system.

2.Curvilinear coordinates56:19

Coordinate points, lines and surfaces on general curvilinear coordinates.

3.Orthogonal curvilinear coordinates31:24

Coordinate points, lines and surfaces on orthogonal curvilinear coordinates.

4.Worked examples (1)41:55

Worked examples on orthogonality, scale factors and Jacobian of transformation of curvilinear coordinate systems.

Orthogonal Curvilinear Coordinates (2)

Meaning of right-handed coordinate systems, vector products of covariant base vectors in right-handed systems, length and volume elements in general orthogonal curvilinear coordinate systems.

Chapter lessons

1.Right-handed systems20:49

Meaning of right-handed systems, and the implication of right-handedness for vector products.

2.Arc length32:58

Arc length (length element) in orthogonal curvilinear coordinates.

3.Volume element57:40

Volume element (elemental volume) in orthogonal curvilinear coordinates and its relation to the Jacobian of transformation.

4.Worked examples (1)33:21

Worked examples length and volume elements in orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (3)

The gradient operator and contravariant bases vectors in general orthogonal curvilinear coordinates.

Chapter lessons

1.Gradient27:38

The gradient of a scalar in orthogonal curvilinear coordinates.

2.Base vectors35:41

Contravariant base vectors in orthogonal curvilinear coordinates, their relation to the covariant base vectors, and their vector product identities.

3.Worked examples (1)42:03

Worked examples on the gradient in orthogonal curvilinear coordinates.

4.Worked examples (2)22:50

More worked examples on the gradient in orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (4)

Divergence and curl operators in general orthogonal curvilinear coordinates.

Chapter lessons

1.Some identities33:49

Some vector product identities.

2.Divergence28:52

The divergence of a vector field in orthogonal curvilinear coordinates.

3.Curl49:34

The curl of a vector field in orthogonal curvilinear coordinates.

4.Worked examples (1)30:32

Worked examples on the divergence and curl of a vector field in orthogonal curvilinear coordinates.

5.Worked examples (2)28:01

Worked examples on the divergence and curl of a vector field in orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (5)

The Laplacian operator in general orthogonal curvilinear coordinates.

Chapter lessons

1.Laplacian (1)16:55

The Laplacian of a scalar field in orthogonal curvilinear coordinates.

2.Laplacian (2)10:22

The Laplacian of a vector field in orthogonal curvilinear coordinates.

3.Worked examples (1)37:01

Worked examples on the Laplacian of scalar and vector fields in orthogonal curvilinear coordinates.

4.Worked examples (2)43:11

More worked examples on the Laplacian of scalar and vector fields in orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (6)

Some special curvilinear coordinates - cylindrical and spherical coordinate systems.

Chapter lessons

1.Cylindrical coordinates30:22

Vector calculus properties in cylindrical coordinates.

2.Spherical coordinates33:34

Vector calculus properties in spherical coordinates.