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Section 17.1

Section 17.1. Vector Fields. VECTOR FIELDS. Definition : Let D be a set in (a plane region). A vector field on is a (vector-valued) function F that assigns to each point ( x , y ) in D a two-dimensional vector F ( x , y ).

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Section 17.1

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  1. Section 17.1 Vector Fields

  2. VECTOR FIELDS Definition: Let D be a set in (a plane region). A vector field on is a (vector-valued) function F that assigns to each point (x, y) in D a two-dimensional vector F(x, y). Since F(x, y) is a two-dimensional vector, we can write it in terms of its component functionsP and Q as follows: The functions P and Q are sometimes called scalar functions.

  3. EXAMPLES Sketch the following vector fields. 1. F(x, y) = −yi + xj 2. F(x, y) = 3xi+ yj

  4. VECTOR FIELDS (CONCLUDED) Definition: Let E be a set in . A vector field on is a function F that assigns to each point (x, y, z) in E a three-dimensional vector F(x, y, z). We can express F in terms of its component functions P, Q, and R as F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k Three dimensional vector fields can be sketched in space. See Figures 9 through 12 on page 1094.

  5. PHYSICAL EXAMPLES OF VECTOR FIELDS • Velocity fields describe the motions of systems of particles in the plane or space. • Gravitational fields are described by Newton’s Law of Gravitation, which states that the force of attraction exerted on a particle of mass m located at x = by a particle of mass M located at (0, 0, 0) is given by where G is the gravitational constant.

  6. PHYSICAL EXAMPLES OF VECTOR FIELDS (CONTINUED) • Electric force fields are defined by Coulomb’s Law, which states that the force exerted on a particle with electric charge q located at (x,y,z) by a particle of charge Q located at (0,0, 0) is given by where x = xi + yj + zk and εis a constant that depends on the units for |x|, q, and Q.

  7. INVERSE SQUARE FIELDS Let x(t) = x(t)i + y(t)j + z(t)k be the position vector. The vector field F is an inverse square field if where k is a real number. Gravitational fields and electric force fields are two physical examples of inverse square fields.

  8. GRADIENTS ANDVECTOR FIELDS Recall that the gradient of a function f (x, y, z) is a vector given by Thus, the gradient is an example of a vector field and is called a gradient vector field. NOTE: There is an analogous gradient field for two dimensions.

  9. CONSERVATIVE VECTOR FIELDS Definition: A vector field F is called a conservative vector field if it is the gradient of some scalar function f , that is if there exists a differentiable function f such that . The function f is called the potential function for F.

  10. EXAMPLES 1. Show that F(x, y) = 2xi + yj is conservative. 2. Show that any inverse square field is conservative.

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