
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 j~Fj = 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 = j~xj. 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 j~xj3 ~x origin and the unit vector in that direction is − ). j~xj 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. 4.
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