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Chapter 16 – Vector Calculus 16.1 Vector Fields Objectives: Understand the different types of vector fields Dr. Erickson 16.1 Vector Fields 1 Vector Calculus In this chapter, we study the calculus of vector fields. ◦ These are functions that assign vectors to points in space. We will be discussing ◦ Line integrals—which can be used to find the work done by a force field in moving an object along a curve. ◦ Surface integrals—which can be used to find the rate of fluid flow across a surface. Dr. Erickson 16.1 Vector Fields 2 Connections The connections between these new types of integrals and the single, double, and triple integrals we have already met are given by the higher-dimensional versions of the Fundamental Theorem of Calculus: ◦ Green’s Theorem ◦ Stokes’ Theorem ◦ Divergence Theorem Dr. Erickson 16.1 Vector Fields 3 Velocity Vector Fields Some examples of velocity vector fields are: ◦ Air currents ◦ Ocean currents ◦ Flow past an airfoil Dr. Erickson 16.1 Vector Fields 4 Force Field Another type of vector field, called a force field, associates a force vector with each point in a region. Dr. Erickson 16.1 Vector Fields 5 Definition – Vector field on 2 Let D be a set in 2 (a plane region). A vector field on 2 is a function F that assigns to each point (x, y) in D a two-dimensional (2-D) vector F(x, y). Dr. Erickson 16.1 Vector Fields 6 Vector Fields on 2 Since F(x, y) is a 2-D vector, we can write it in terms of its component functions P and Q as: F(x, y) = P(x, y) i + Q(x, y) j = <P(x, y), Q(x, y)> or, for short, F=Pi+Qj Dr. Erickson 16.1 Vector Fields 7 Definition - Vector Field on 3 Let E be a subset of 3. A vector field on 3 is a function F that assigns to each point (x, y, z) in E a three-dimensional (3-D) vector F(x, y, z). Dr. Erickson 16.1 Vector Fields 8 Velocity Fields Imagine a fluid flowing steadily along a pipe and let V(x, y, z) be the velocity vector at a point (x, y, z). ◦ Then, V assigns a vector to each point (x, y, z) in a certain domain E (the interior of the pipe). ◦ So, V is a vector field on 3 called a velocity field. Dr. Erickson 16.1 Vector Fields 9 Velocity Fields A possible velocity field is illustrated here. ◦ The speed at any given point is indicated by the length of the arrow. Dr. Erickson 16.1 Vector Fields 10 Gravitational Fields The gravitational force acting on the object at x = <x, y, z> is: mMG F ( x) x 3 |x| Note: Physicists often use the notation r instead of x for the position vector. So, you may see Formula 3 written in the form F = –(mMG/r3)r Dr. Erickson 16.1 Vector Fields 11 Electric Fields Instead of considering the electric force F, physicists often consider the force per unit charge: 1 Q E(x) F(x) 3 x q |x| ◦ Then, E is a vector field on 3 called the electric field of Q. Dr. Erickson 16.1 Vector Fields 12 Example 1 Match the vector fields F with the plots labeled I-IV. Give reasons for your choices. 1. F(x, y) = <1, siny> 1. F(x, y) = <y, 1/x> Dr. Erickson 16.1 Vector Fields 13 Example 2 – pg. 1086 # 34 At time t = 1, a particle is located at position (1, 3). If it moves in a velocity field F x, y xy 2, y 2 10 find its approximate location at t = 1.05. Dr. Erickson 16.1 Vector Fields 14