Vector Calculus Part 1: Gradient Divergence & Curl of vector

Gradient , Divergence and Curl of a Vector, Tangent and Normal plane, vector calculus, Laplacian operator, Del Operator

Created byJaswinder Kaur

Last updated 2/2025

English

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What you'll learn

Basic concepts of Vectors
Scalar and Vector point functions
Directional Derivatives
Geometrical interpretation of vectors
Geometrical Interpretation of Gradient of scalar point function
Gradient
Divergence and Curl of a vector
Divergence and Curl of a vector
Del Operator
Laplacian operator
Solenoidal vector
Irrotational vector
Important various results and Expected Theorems and based Assignments

Course content

3 sections • 26 lectures • 7h 59m total length

Introduction13:41
Concept of Scalar and Vector Function , Scalar function of a scalar variable , Vector function of a scalar variable , Single Valued Function , Limit of a Vector Function , Algebra of limits , Variable Function , Diffrentiability of the Vector Function
Basic Concepts Lesson 216:42
Some Important Results and Theorems
Theorem : Every Diffrentiable Vector Function is Continous.
Constant Vectors Lesson 118:55
Definition of Constant vector and some Important Theorems.
Constant Vectors Lesson 213:33
Introduction of a Null Vector and Important Theorem.
Constant Vectors Lesson 322:10
Some more Theorems and Geometrical Significance of df/dt .

Geometrical Significance of Scalar And Vector Point Function10:36
Geometrical Significance of Velocity and Accelaration
Scalar and Vector Point Functions
Limits and Continuity
Directional Derivatives19:35
Cartesian Representation of Point Function and their Directional Derivatives.
Directional Derivative of Scalar Point Function along Coordinate Axes.
Directional Derivative along any Line.
Gradient of a Scalar Point Function. and Gradient of a Constant.
Introduction of Del Operator18:17
Important Expected Theorem and some Examples.
Geometrical Interpretation of a Gradient of a Scalar Point Function25:31
Basic concept of Direction Cosines and Level Surfaces.
Greometrical Interpretation of a Gradient of a Scalar Point Function.
Gradient of a Scalar Point Function ( Examples )24:38
Explore gradient, directional derivatives, and unit normals to level surfaces through practical examples, illustrating greatest rate of increase and how to compute with dot and cross products.
Tangent Plane and Normal14:11
Conditions of Gradient for Tangent Plane and Normal and solved Exercise.
Divergence and Curl of a Vector22:50
Explore divergence and curl of vector fields using the del operator, dot and cross products, and determinant forms, with worked examples and practice.

Assignment 138:56
Explore solenoidal and rotational vector fields, with emphasis on divergence, curl, and the Laplacian. Learn vector calculus identities involving F and G, and cross products with the position vector.
Assignment 220:33
Explore gradient, divergence, and curl for three-dimensional vector fields, solving assignment 2: show ∇·(r/r^3) = 0 (r ≠ 0) and ∇·â_r = 2/r using determinant methods.
Assignment 313:37
Explore assignment three in vector calculus, proving grad(1/r) = -r/r^3, and showing for a constant vector A that div(A × r) = 0 and curl(A × r) = 2A.
Assignment 418:42
Solve assignment 4 on vector calculus, applying gradient, divergence, and curl concepts through cross and dot products and determinant methods, verifying zero divergence cases and zero curl conditions.
Assignment 537:23
Assignment 5 surveys key remarks and formulas in vector calculus part 1, detailing gradient, divergence, and curl, and develops dot and cross product rules for differentiable vector and scalar functions.
Assignment 618:32
Assignment six examples illustrate the product rule for divergence with gradient fields, derive divergence of gradient, and prove that the curl cross product yields solenoids; compute divergence and curl steps.
Assignment 715:16
Establish the necessary and sufficient condition for a vector to be a gradient: its curl must be zero. A zero curl implies a gradient, with divergence considerations.
Assignment 816:00
Assignment eight demonstrates that the divergence of a cross product is zero, using gradient and curl concepts, with examples and practice problems.
Assignment 914:09
Assignment 1013:58
Assignment 1112:17
Assignment 11 guides differentiating with respect to x, y, and z to show that the laplacian of x/r^3 equals zero for r = sqrt(x^2 + y^2 + z^2).
Assignment 1215:24
Assignment 138:56
Derive that the divergence of the gradient of 1/r equals zero away from the origin, using partial derivatives and the gradient, divergence, and Laplacian relationships in vector calculus.
Assignment 1415:05
Assignment 14 develops a proof relating gradient and divergence through x, y, z derivatives, comparing right and left hand sides and exploring bisymmetry to establish the identity.

Requirements

Vectors
Unit vector
Direction cosines
Partial Derivatives
Cross Product of Vectors
Dot Product of vectors

Description

In course , Vector Calculus Part 1 the student will learn about the following topics:

Basic concepts of Vectors and detailed definitions
Scalar and Vector point functions
Constant Vectors and all Based Theorems.
The proof of the Theorem that every Differentiable vector function is Continuous.
The proof of the result that_The Necessary and sufficient condition for a vector point function to be Constant.
The proof of the result that _If vector function has a Constant magnitude then f and df/dt are perpendicular.
The Necessary and sufficient condition for a vector function to have Constant magnitude.
The Necessary and sufficient condition for a vector function to have Constant direction.
Directional Derivatives with examples
Tangent Plane and Normal, Level Surfaces with definitions and detailed explanation
Geometrical interpretation of vectors
Geometrical Interpretation of Gradient of scalar point function
Gradient, Divergence, and Curl of a vector and Many more Based examples and assignments with Theorems and proofs.
Del Operator & Laplacian operator with examples.
Solenoidal vector & Irrotational vector
Important various Results, Expected Theorems, and Based Assignment

If you need any help in understanding the topics or If you have any queries, feel free to revert back. The instructor is always there to help you.

Thanks and Regards!

Who this course is for:

Graduates
Post Graduates
Engineering students
Physics Students
Students of Mathematical Sciences

Vector Calculus Part 1: Gradient Divergence & Curl of vector

What you'll learn

Explore related topics

Course content

Basic Concepts of Vector Calculus5 lectures • 1hr 25min

Directional Derivatives , GRADIENT , DIVERGENCE AND CURL of a vector.7 lectures • 2hr 16min

Assignments on Solenoidal Vector , Irrotational Vector , Laplacian Operator.14 lectures • 4hr 19min

Requirements

Description

Who this course is for: