Math 2415 – Calculus III Section 16.1 Vector Fields

• The rotation of a hurricane is an example of a vector field.

• Before, we looked at vector functions whose domains were sets of real numbers and whose ranges were sets of vectors. Now we will discuss vector fields.

• Definition: Let D be a set in (a plane region). A is a function ~F that assigns to each point (x,y) in D a two-dimensional vector ~F(x,y).

• The best way to visualize a vector field is to start at the point (x,y) and draw the arrow representing ~F(x,y) from that point.

• ~F(x,y) =

• Definition: Let E be a subset of .A is a function ~F that assigns to each point (x,y,z) in E a three-dimensional vector ~F(x,y,z).

• ~F(x,y,z) =

• We may write ~F(~x) instead of ~F(x,y,z). (Represent the point (x,y,z) with its position vector ~x =< x,y,z >). Then ~F is a function that assigns the vector ~F(~x) to a vector~x. Math 2415 Section 16.1 Continued

Ex: Sketch some vectors in the vector field given by ~F(x,y) = −yˆı + x jˆ.

Ex: Sketch some vectors in the vector field given by ~F(x,y) = 2xˆı + y jˆ.

2 Math 2415 Section 16.1 Continued

• Physical examples:

1. Velocity Fields Describe motion of particles in the plane or space. (Example: a wheel rotating) – Velocity vectors are determined by

– Velocity fields are also determined by the flow of liquids through a container or the flow of air currents around a moving object. 2. Gravitational Fields Defined by Newton’s Law of Gravitation. mMG |~F| = relates the magnitude of the gravitational force between two objects with masses M and m r2 with the distance between the objects r, where G is the gravitational constant.

3 If we assume the object with mass M is at the origin of R and let ~x =< x,y,z > be the position vector of the object with mass m, then r = |~x|. We can write a formula for the gravitational force acting on the mMG second object at~x as ~F(~x) = − ~x (since the gravitational force of the second object acts toward the |~x|3 ~x origin and the unit vector in that direction is − ). |~x| The formula may also be written as −mMGx −mMGy −mMGz ~F(x,y,z) = iˆ+ jˆ+ kˆ (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2

3 Ex: Sketch the vector field on R given by ~F(x,y,z) = zkˆ.

3 Math 2415 Section 16.1 Continued

• Gradient Fields

~ ~ 2 Recall: ∇ f (x,y) =< fx(x,y), fy(x,y) >. Thus, ∇ f is a vector field on R , a gradient vector field. 3 • Likewise, we have the gradient vector field on R : ~ ∇ f (x,y,z) =< fx(x,y,z), fy(x,y,z), fz(x,y,z) >. 2 Ex: Find the gradient vector field of f (x,y,z) = z − yex .

• Definition: A vector field ~F is called a conservative vector field if it is the gradient of some scalar function (i.e. ) f is a for ~F.

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