Vector Field Circle

Given a subset S of R n, a vector field is represented by a vector-valued function V S R n in standard Cartesian coordinates x 1, , x n.If each component of V is continuous, then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function differentiable any number of times. A vector field can be visualized as

a vector field in which the vector at point 92x,y92 is tangent to a circle with radius 92r92sqrtx2y292 in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin unit vector field a vector field in which the magnitude of every vector is 1

In contrast to radial fields, in a rotational field, the vector at point latexx, ylatex is tangent not perpendicular to a circle with radius latexr92sqrtx2y2latex. In a standard rotational field, all vectors point either in a clockwise direction or in a counterclockwise direction, and the magnitude of a vector depends only on

A vector field is the compilation of these vectors at every point. We draw vector field with evenly spread points for visual purposes, but you should imagine the field as a continuum. A vector field by itself has no meaning, but for the purpose of this section, we will call the field 92F92 because force is a common use of the vector field.

A vector field 9292vec F92 is called a conservative vector field if there exists a function 92f92 such that 9292vec F 92nabla f92. If 9292vec F92 is a conservative vector field then the function, 92f92, is called a potential function for 9292vec F92. All this definition is saying is that a vector field is conservative if it is also a gradient

16.1.1 What is a Vector Field? A vector field is a function of several variables that assigns a each point in its domain to a vector. Formally, Definition 92mathbfF92mathbfx is perpendicular to 92mathbfx and hence tangent to the circle centered at the origin with radius 92mathbfx 92sqrtx2 y2.

and then show that it is the flow of the vector field F x,y -y, x .. Solution It is easy to show that r 2 x 2 y 2, thus implying that is a system of concentric circles centered at 0,0 .Motion on each circle is uniform circular motion, as is shown below along with the vector field.

A vector field on the circle is a simple enough object. In this chapter, vector fields are considered in relation to diffeomorphisms. This chapter reviews vector fields with zeros of finite order and presents a complete set of invariants for such vector fields. Some are locally determined at the zeros and others are global.

Vector . Fields page . 554 CHAPTER . 15 . VECTOR CALCULUS . 15.1 Vector . Fields page 554 A . vector field assigns avector to each point x, y or x, y, The work is 18r around the complete circle. Formally gz, yds is the limit of the sum gq,yiAsi. The four equivalent properties of a conservative field F

be the unit circle with counterclockwise orientation. Find the circulation on . C. for the following vector fields. a. T h e r a d i a l f l o w f i e l d. F the vector field is in the direction of the tangent vector the result is a positive circulation Figure 17.24.