Vector calculus | Differential topology
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to 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 strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow). In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field. (Wikipedia).
Introduction to Vector Fields This video discusses, 1) The definition of a vector field. 2) Examples of vector fields including the gradient, and various velocity fields. 3) The definition of a conservative vector field. 4) The definition of a potential function. 5) Test for conservative
From playlist Calculus 3
11_7_1 Potential Function of a Vector Field Part 1
The gradient of a function is a vector. n-Dimensional space can be filled up with countless vectors as values as inserted into a gradient function. This is then referred to as a vector field. Some vector fields have potential functions. In this video we start to look at how to calculat
From playlist Advanced Calculus / Multivariable Calculus
This video explains the definition of a vector space and provides examples of vector spaces.
From playlist Vector Spaces
Calculus 3 Lecture 15.1: INTRODUCTION to Vector Fields (and what makes them Conservative)
Calculus 3 Lecture 15.1: INTRODUCTION to Vector Fields (and what makes them Conservative): What Vector Fields are, and what they look like. We discuss graphing Vector Fields in 2-D and 3-D and talk about what a Conservative Vector Field means.
From playlist Calculus 3 (Full Length Videos)
Worldwide Calculus: Vector Fields
Lecture on 'Vector Fields' from 'Worldwide Multivariable Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.
From playlist Integration and Vector Fields
Intro to VECTOR FIELDS // Sketching by hand & with computers
Vector Fields are extremely important in math, physics, engineering, and many other fields. Gravitational fields, electric fields, magnetic fields, velocity fields, these are all examples of vector fields. In this video we will define the concept of a vector field, talk about some basic te
MATH2018 Lecture 3.2 Vector Fields
We discuss the concept of a vector field and introduce some basic tools for understanding them: divergence and curl.
From playlist MATH2018 Engineering Mathematics 2D
Free ebook http://tinyurl.com/EngMathYT A basic introduction to vector fields discussing the need for vector fields and some of the basic mathematics associated with them.
From playlist Engineering Mathematics
Brent Pym: Holomorphic Poisson structures - lecture 2
The notion of a Poisson manifold originated in mathematical physics, where it is used to describe the equations of motion of classical mechanical systems, but it is nowadays connected with many different parts of mathematics. A key feature of any Poisson manifold is that it carries a cano
From playlist Virtual Conference
What is a vector field?? Chris Tisdell UNSW
This lecture gently introduces the idea of a vector field. Dr Chris Tisdell discusses the need for a vector field, plus presents many examples. This lecture is from a course of Vector Calculus, which is taught at UNSW, Sydney by Dr Chris Tisdell..
From playlist Vector Calculus @ UNSW Sydney. Dr Chris Tisdell
What is a Tensor? Lesson 21: The Lie derivative
What is a Tensor? Lesson 21: The Lie derivative We reconstruct the notion of a vector space at a point in spacetime using the more fundamental exposition of tangent vectors to curves. Then we define a congruence of curves associated with a vector field and then we define the Lie derivativ
From playlist What is a Tensor?
Adolfo Guillot: Complete holomorphic vector fields and their singular points - lecture 2
CIRM VIRTUAL EVENT Recorded during the research school "Geometry and Dynamics of Foliations " the May 11, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on C
From playlist Virtual Conference
Worldwide Calculus: Conservative Vector Fields
Lecture on 'Conservative Vector Fields' from 'Worldwide Multivariable Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.
From playlist Integration and Vector Fields
What is a Tensor? Lesson 18: The covariant derivative continued
What is a Tensor? Lesson 18: The covariant derivative continued This lesson covers some of the "coordinate free" language used to describe the covariant derivative. As a whole this lecture is optional. However, becoming comfortable with coordinate free language is probably a good idea. I
From playlist What is a Tensor?
What is a Vector Space? Definition of a Vector space.
From playlist Linear Algebra
The Curl of a Vector Field: Measuring Rotation
This video introduces the curl operator from vector calculus, which takes a vector field (like the fluid flow of air in a room) and returns a vector field quantifying how much, and about what direction, the flow is locally rotating, or swirling, at every point. The curl is a fundamental b
From playlist Engineering Math: Vector Calculus and Partial Differential Equations