Vector Calculus

This course deals with vector calculus and its applications. The first section explains the line integrals and vector fields. The course progresses with the introduction of conservative fields and potential functions, Green's theorem. The next sections deal with surface integral, Stokes theorem, Divergence theorem and their application in real life.

10 topics

121 lectures

Educators

Course Curriculum

6 videos

The Dot Product

5 videos

The Cross Product

5 videos

Lines and Planes in Space

5 videos

Vector Functions

16 videos

Functions of Several Variables

11 videos

Partial Derivatives

10 videos

Multivariable Optimization

16 videos

Multiple Integrals

21 videos

Vector Calculus

26 videos

Vector Calculus Lectures

Vector Calculus

Vector Fields - Intro

In mathematics, a vector field is a mathematical tool used to describe the velocity and acceleration of particles in a fluid, the speed and direction of a moving object, the distribution of heat over a surface, or the strength and direction of some physical force. A vector field in three-dimensional space is represented by a line segment, called a vector, that has a definite magnitude and direction at each point. The vector is commonly depicted by a line with an arrow, as in the diagram at right. The vector itself is often referred to as the "field". The magnitude of the vector is called the "length" of the vector, and the direction is given by the angle the vector makes with the positive "x"-axis. The vector is said to be "normalized" if its length is equal to 1. The scalar (or dot) product of two vectors is a numerical value that determines the angle between them. Dot products are used to calculate the magnitude of a vector or the angle between two vectors.

Matt Just

Vector Calculus

Line Integrals and Vector Fields - Overview

In mathematics, a vector field is a map which assigns a vector to each point in a subset of space. A vector field in the plane, can be visualized as a collection of arrows with a given magnitude and direction, each arrow being associated with a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the amount of some substance in a blood vessel at each point in space.

Matt Just

Vector Calculus

Line Integrals and Vector Fields - Example 1

In mathematics, a vector field is a map which assigns a vector to each point in a subset of space. A vector field in the plane, can be visualized as a collection of arrows with a given magnitude and direction, each arrow being associated with a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the amount of some substance in a blood vessel at each point in space.

Matt Just

Vector Calculus

Line Integrals and Vector Fields - Example 2

In mathematics, a vector field is a map which assigns a vector to each point in a subset of space. A vector field in the plane, can be visualized as a collection of arrows with a given magnitude and direction, each arrow being associated with a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the amount of some substance in a blood vessel at each point in space.

Matt Just

Vector Calculus

Line Integrals and Vector Fields - Example 3

In mathematics, a vector field is a map which assigns a vector to each point in a subset of space. A vector field in the plane, can be visualized as a collection of arrows with a given magnitude and direction, each arrow being associated with a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the amount of some substance in a blood vessel at each point in space.

Matt Just

Vector Calculus

Line Integrals and Vector Fields - Example 4

In mathematics, a vector field is a map which assigns a vector to each point in a subset of space. A vector field in the plane, can be visualized as a collection of arrows with a given magnitude and direction, each arrow being associated with a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the amount of some substance in a blood vessel at each point in space.

Matt Just

Vector Calculus

Conservative Fields and Potential Functions - Overview

In physics, a conservative field is a field that satisfies the following condition: if we have a small volume that is infinitesimally small, the flux of the field through that volume is the same as if the field were constant over that volume.

Matt Just

Vector Calculus

Conservative Fields and Potential Functions - Example 1

In physics, a conservative field is a field that satisfies the following condition: if we have a small volume that is infinitesimally small, the flux of the field through that volume is the same as if the field were constant over that volume.

Matt Just

Vector Calculus

Conservative Fields and Potential Functions - Example 2

In physics, a conservative field is a field that satisfies the following condition: if we have a small volume that is infinitesimally small, the flux of the field through that volume is the same as if the field were constant over that volume.

Matt Just

Vector Calculus

Conservative Fields and Potential Functions - Example 3

In physics, a conservative field is a field that satisfies the following condition: if we have a small volume that is infinitesimally small, the flux of the field through that volume is the same as if the field were constant over that volume.

Matt Just

Vector Calculus

Conservative Fields and Potential Functions - Example 4

In physics, a conservative field is a field that satisfies the following condition: if we have a small volume that is infinitesimally small, the flux of the field through that volume is the same as if the field were constant over that volume.

Matt Just

Vector Calculus

Green's Theorem - Overview

In mathematics, Green's theorem, also known as the divergence theorem or the fundamental theorem of calculus, is a theorem in calculus in which the integral of a function over an arbitrary region in the plane is found by computing the line integral around any closed curve that intersects the region. The theorem is named after George Green, who stated and proved it in 1828.

Matt Just

Vector Calculus

Green's Theorem - Example 1

In mathematics, Green's theorem, also known as the divergence theorem or the fundamental theorem of calculus, is a theorem in calculus in which the integral of a function over an arbitrary region in the plane is found by computing the line integral around any closed curve that intersects the region. The theorem is named after George Green, who stated and proved it in 1828.

Matt Just

Vector Calculus

Green's Theorem - Example 2

In mathematics, Green's theorem, also known as the divergence theorem or the fundamental theorem of calculus, is a theorem in calculus in which the integral of a function over an arbitrary region in the plane is found by computing the line integral around any closed curve that intersects the region. The theorem is named after George Green, who stated and proved it in 1828.

Matt Just

Vector Calculus

Green's Theorem - Example 3

In mathematics, Green's theorem, also known as the divergence theorem or the fundamental theorem of calculus, is a theorem in calculus in which the integral of a function over an arbitrary region in the plane is found by computing the line integral around any closed curve that intersects the region. The theorem is named after George Green, who stated and proved it in 1828.

Matt Just

Vector Calculus

Green's Theorem - Example 4

In mathematics, Green's theorem, also known as the divergence theorem or the fundamental theorem of calculus, is a theorem in calculus in which the integral of a function over an arbitrary region in the plane is found by computing the line integral around any closed curve that intersects the region. The theorem is named after George Green, who stated and proved it in 1828.

Matt Just

Vector Calculus

Surface Integrals - Overview

In mathematics, a surface integral is an integral along a surface in three-dimensional Euclidean space. It is a generalization of the line integral to surfaces of arbitrary genus.

Matt Just

Vector Calculus

Surface Integrals - Example 1

In mathematics, a surface integral is an integral along a surface in three-dimensional Euclidean space. It is a generalization of the line integral to surfaces of arbitrary genus.

Matt Just

Vector Calculus

Surface Integrals - Example 2

In mathematics, a surface integral is an integral along a surface in three-dimensional Euclidean space. It is a generalization of the line integral to surfaces of arbitrary genus.

Matt Just

Vector Calculus

Surface Integrals - Example 3

In mathematics, a surface integral is an integral along a surface in three-dimensional Euclidean space. It is a generalization of the line integral to surfaces of arbitrary genus.

Matt Just

Vector Calculus

Surface Integrals - Example 4

In mathematics, a surface integral is an integral along a surface in three-dimensional Euclidean space. It is a generalization of the line integral to surfaces of arbitrary genus.

Matt Just

Vector Calculus

Stokes' Theorm and Divergence Theorem - Overview

In mathematics, the divergence theorem is a theorem about vector fields. It states that the divergence of a vector field is zero in a region if and only if the field is the gradient of a scalar field. The theorem is named for the mathematician George Green, who stated it in 1828. The theorem is also known as the Kelvin–Stokes theorem, after William Thomson, 1st Baron Kelvin (Lord Kelvin) and George Gabriel Stokes.

Matt Just

Vector Calculus

Stokes' Theorm and Divergence Theorem - Example 1

In mathematics, the divergence theorem is a theorem about vector fields. It states that the divergence of a vector field is zero in a region if and only if the field is the gradient of a scalar field. The theorem is named for the mathematician George Green, who stated it in 1828. The theorem is also known as the Kelvin–Stokes theorem, after William Thomson, 1st Baron Kelvin (Lord Kelvin) and George Gabriel Stokes.

Matt Just

Vector Calculus

Stokes' Theorm and Divergence Theorem - Example 2

In mathematics, the divergence theorem is a theorem about vector fields. It states that the divergence of a vector field is zero in a region if and only if the field is the gradient of a scalar field. The theorem is named for the mathematician George Green, who stated it in 1828. The theorem is also known as the Kelvin–Stokes theorem, after William Thomson, 1st Baron Kelvin (Lord Kelvin) and George Gabriel Stokes.

Matt Just

Vector Calculus

Stokes' Theorm and Divergence Theorem - Example 3

In mathematics, the divergence theorem is a theorem about vector fields. It states that the divergence of a vector field is zero in a region if and only if the field is the gradient of a scalar field. The theorem is named for the mathematician George Green, who stated it in 1828. The theorem is also known as the Kelvin–Stokes theorem, after William Thomson, 1st Baron Kelvin (Lord Kelvin) and George Gabriel Stokes.

Matt Just

Vector Calculus

Stokes' Theorm and Divergence Theorem - Example 4

In mathematics, the divergence theorem is a theorem about vector fields. It states that the divergence of a vector field is zero in a region if and only if the field is the gradient of a scalar field. The theorem is named for the mathematician George Green, who stated it in 1828. The theorem is also known as the Kelvin–Stokes theorem, after William Thomson, 1st Baron Kelvin (Lord Kelvin) and George Gabriel Stokes.

Matt Just