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Chapter 16 – Vector Calculus

Chapter 16 – Vector Calculus. 16.1 Vector Fields. Objectives: Understand the different types of vector fields. 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

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Chapter 16 – Vector Calculus

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  1. Chapter 16 – Vector Calculus 16.1 Vector Fields • Objectives: • Understand the different types of vector fields 16.1 Vector Fields

  2. 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. 16.1 Vector Fields

  3. 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 16.1 Vector Fields

  4. Velocity Vector Fields • Some examples of velocity vector fields are: • Air currents • Ocean currents • Flow past an airfoil 16.1 Vector Fields

  5. Force Field • Another type of vector field, called a force field, associates a force vector with each point in a region. 16.1 Vector Fields

  6. Definition – Vector field on 2 • Let D be a set in 2(a plane region). • A vector field on 2is a function F that assigns to each point (x, y) in D a two-dimensional (2-D) vector F(x, y). 16.1 Vector Fields

  7. Vector Fields on 2 • Since F(x, y) is a 2-D vector, we can write it in terms of its component functions Pand 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 16.1 Vector Fields

  8. Definition - Vector Field on 3 • Let E be a subset of 3. • A vector field on 3is a function F that assigns to each point (x, y, z) in E a three-dimensional (3-D) vector F(x, y, z). 16.1 Vector Fields

  9. 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,Vassigns 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. 16.1 Vector Fields

  10. Velocity Fields • A possible velocity field is illustrated here. • The speed at any given point is indicated by the length of the arrow. 16.1 Vector Fields

  11. Gravitational Fields • The gravitational force acting on the object at x = <x, y, z>is: • Note: Physicists often use the notation rinstead of x for the position vector. So, you may see Formula 3 written in the form F = –(mMG/r3)r 16.1 Vector Fields

  12. Electric Fields • Instead of considering the electric force F, physicists often consider the force per unit charge: • Then, E is a vector field on 3 called the electric fieldof Q. 16.1 Vector Fields

  13. 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> 16.1 Vector Fields

  14. Example 2 – pg. 1086 # 34 • At time t = 1, a particle is located at position (1, 3). If it moves in a velocity field find its approximate location at t = 1.05. 16.1 Vector Fields

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