Worksheet 4.2: Introduction to Vector Fields and Line Integrals

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1 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals From the Toolbox (what you need from previous classes) Know what a vector is. Be able to sketch vectors. Be able to evaluate a scalar-valued function f (x, y). Know what a tangent vector to a curve is, and be able to sketch one on a diagram. Be able to use a diagram representing two vectors to determine whether their dot product is positive, negative, or zero, based on the angle between them. Be able to compute a gradient f (x, y). Goals In this worksheet, you will: Plot vector fields by hand and using graphing apps. Match the plot of a vector field with its defining equation. Recognize examples of important vector fields: radial fields, rotational fields, and gradient fields. Determine whether a vector field is a gradient field by using the equation of the vector field to find the equation of a potential function. Explore how a vector fields interacts with the tangent vector to a curve using the dot product. Definition: Vector Fields A vector field is a function where an input is a point, and an output is a vector.

2 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 1 Model 1: Sketching Vector Fields DIAGRAM 1: Diagram 1 shows four vectors of the vector field: F (x, y) = (x + y) î + 3 ĵ Critical Thinking Questions In this section, you will practice evaluating and sketching vector fields. (Q1) Complete the table of component forms and initial points for the four vectors pictured in Diagram 1. Vector = Component Form Initial Point F (3, 2) = 5 î + 3 ĵ ( ) 3, 2 F ( 3, 2) = î + ĵ (, ) F ( 3, 2) = î + ĵ (, ) F (3, 2) = î + ĵ (, )

3 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 2 (Q2) Complete this statement: The initial point of the vector F (a, b) is the point. (Q3) Compute the component form of F (2, 5), then sketch it on Diagram 1 at its initial point (2, 5). (Q4) Compute the component form of F (5, 7), then sketch it on Diagram 1. (Q5) Compute the component form of F (0, 0), then sketch it on Diagram 1. (Q6) Compute the component form of F ( 5, 5), then sketch it on Diagram 1. (Q7) Is the following statement true, or false? If it is false, give an example that shows it is false. The x-component of the vectors of the vector field F (x, y) = (x + y) î + 3 ĵ is always positive. (Q8) Explain why the following statement about the vector field F (x, y) = (x + y) î + 3 ĵ is false: There is some point (a, b) so that the y-component of F (a, b) equals zero. ( Q9) Write the equation of a vector field G(x, y) such that the x-component of G(x, y) is always positive, and the y-component is always negative.

4 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 3 Model 2: Examples of Vector Fields DIAGRAM 2: Vector Fields Vector Field A Vector Field B Vector Field C Vector Field D F 1 (x, y) = x î + y ĵ EQUATIONS OF VECTOR FIELDS: F 2 (x, y) = y î + x ĵ F 3 (x, y) = y î + x ĵ F 4 (x, y) = (x y) î + (x + y) ĵ Critical Thinking Questions In this section, you will work with a few important examples of vector fields.

5 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 4 (Q10) Match each of the vector field plots in Diagram 2 with its equation from the bottom of Model 2. You may use a graphing app for example: (Q11) A radial field is a vector field where either all vectors point directly towards the origin, or all vectors point directly away from the origin (so, in R 2 all vectors are orthogonal to circles centered at the origin, and in R 3, all vectors are orthogonal to spheres centered at the origin). Which (if any) of the vector fields in Model 2 is/are radial fields? (Q12) A spin field (or rotational vector field) in R 2 is a vector field where all vectors are tangent to circles centered at the origin, and either all vectors point clockwise, or all vectors point counterclockwise. Which (if any) of the vector fields in Model 2 is/are spin fields? (Q13) A constant field is a vector field where all vectors have the same direction, and the same magnitude. Which (if any) of the vector fields in Model 2 is/are constant fields? A gradient field f is a vector field that is the gradient of a multivariate function f. f is called a potential function of the gradient field f. (Q14) Show that f (x, y) = x 2 + 3y is a potential function for f (x, y) = 2x, 3 in two ways: (a) Start with the potential function f. Take the partial derivatives of f with respect to x and y, and show that they are the same as the component functions of the vector field. (b) Start with the gradient field f. Integrate the x-component function of f with respect to x, and the y-coordinate function with respect to y. Compare your results to f.

6 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 5 (Q15) (a) Find a potential function f (x, y) for the gradient field f (x, y) = y + 2x, x + 5 is the gradient of a function f (x, y). (b) Can you find other potential functions for the gradient field in (a)? How many? (c) Which (if any) of the vector fields in Model 2 are gradient fields? What are their potential functions?

7 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 6 Model 3: Vector Fields & Curves DIAGRAM 3: Diagram 3 shows the plot of vector field: F (x, y) = P (x, y) î + Q(x, y) ĵ and a curve C, described by the vector equation: r(t) = x(t) î + y(t) ĵ The points R, S, and T lie on the curve C. The direction of travel along the curve C is from left to right (indicated by the arrow.) You don t know the equation for either the field F (x, y) or the vector function r(t).

8 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 7 Critical Thinking Questions In this section, you explore how vector fields interact with tangent vectors to curves, via the dot product. (Q16) On Diagram 3, at each of the points R, S, and T : (a) At R and T, sketch a vector representing the field vector at the point. At the point S, you can assume the field vector is 0. Instead of sketching an arrow, circle the point S. (b) Sketch the vector dr, as viewed under the infinite magnifying glass. (Q17) At the point R (choose one): F dr > 0 F dr < 0 F dr = 0 At the point S (choose one): F dr > 0 F dr < 0 F dr = 0 At the point T (choose one): F dr > 0 F dr < 0 F dr = 0 (Q18) If you compute the dot product F dr at every point on a curve C, and then add up (integrate) F dr for every point on the curve, you have a vector line integral C F dr. Using this idea, estimate the sign of C F dr for the curve C and vector field F (x, y) in Diagram 3 of Model 3. (choose one): dr > 0 dr < 0 C F dr = 0 (Q19) Go back to Model 2, Diagram 2. On each of the four vector field plots, sketch a circle C centered at the origin, oriented counterclockwise. Use the same method as in (Q18) to determine the sign of C F dr for each of these fields, where C is the circle you sketched. Vector Field A (choose one): dr > 0 dr < 0 C F dr = 0 Vector Field B (choose one): dr > 0 dr < 0 C F dr = 0 Vector Field C (choose one): dr > 0 dr < 0 C F dr = 0 Vector Field D (choose one): dr > 0 dr < 0 C F dr = 0 (Q20) Choose one of of the vector fields from (Q19) for which you got a non-zero number for C F dr. What could you do to either the vector field or the curve in order to change the sign of your answer?

9 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 8 (Q21) Below is a plot of the same vector field from Model 3, Diagram 3. (Remember, you do not have an equation that defines this field: you have only the plot to work with.) Sketch curves C 1, C 2, and C 3 so that: ˆ F dr > 0 You travel along the curve C 1, taking the dot product of F and dr at each C 1 point. When you add up all of these dot products, the number you get is positive. ˆ F dr < 0 You travel along the curve C 2, taking the dot product of F and dr at each C 2 point. When you add up all of these dot products, the number you get is negative. ˆ F dr = 0 You travel along the curve C 3, taking the dot product of F and dr at each C 3 point. When you add up all of these dot products, you get zero.

10 Boise State Math 275 (Ultman) Worksheet 4.2: Introduction to Vector Fields and Line Integrals 9 Summary A vector field is a function which assigns an n-dimensional vector to every point in R n, or some subset of R n. In R 2 : F (x, y) = P (x, y) î + Q(x, y) ĵ. When you look at a diagram representing a vector field, you see an arrow based at each point. To determine what the vector is at each point, evaluate the component functions of the field F (x, y, z) at the x, y, and z values at the point. You have already studied one example of vector field: the gradient f of a multivariate function. (Important note: not every vector field is a gradient field.) The tangent vector to a curve passing through a vector field interacts with the field vectors via the dot product. (This is the idea behind vector line integrals.)

Worksheet 1.7: Introduction to Vector Functions - Position

Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position From the Toolbox (what you need from previous classes): Cartesian Coordinates: Coordinates of points in general,