Topic 5.2: Introduction to Vector Fields

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Eugene Gaines
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1 Math 75 Notes Topic 5.: Introduction to Vector Fields Tetbook Section: 16.1 From the Toolbo (what you need from previous classes): Know what a vector is. Be able to sketch a vector using its component form. Be able to evaluate scalar-valued functions f (, y) and f (, y, z). Be able to compute the gradient f of a function f, and be able to find a function f (if such a function eists) that results in a given gradient f. 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. Learning Objectives (New Skills) & Important Concepts Learning Objectives (New Skills): Plot vector fields by hand and using graphing apps. Match the plot of a vector field with its defining equation. Recognize eamples 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.

2 Important Concepts: A vector field is a function whose input is a point P = (, y, z), and whose output is a vector F (, y, z). When you look at a diagram representing a vector field, you see an arrow based at each point. Some important eamples of vector fields are: The gradient field f of a multivariate function f. Radial fields, which radiate from the origin. Rotational fields, which rotate about the origin in -d. (Note: Not every field falls into one of the above categories! These are just some important types of fields you will encounter in this and other classes.) 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.) The Big Picture A vector field is a function whose input is a point P = (, y, z), and whose output is a vector: F (, y, z) = P (, y, z) î + Q(, y, z) ĵ + R(, y, z) ˆk. The component functions P (, y, z), Q(, y, z), R(, y, z) of the vector field are multivariate scalar-valued functions (the type of function we have been studying so far this semester). 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 (, y, z) at the, y, and z values at the point. Vector fields are important because many physical phenomena can be represented as vector fields. For eample: velocity fields, gravitational fields, electric fields, and magnetic fields can all be represented by vector fields.

3 More Details 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 : In R 3 : F (, y) = P (, y) î + Q(, y) ĵ F (, y, z) = P (, y, z) î + Q(, y, z) ĵ + R(, y, z) ˆk The functions P, Q, R are called component functions. They are multivariate scalar-valued functions (the type of function we have been studying so far this semester). F is continuous if its component functions are continuous. F is differentiable if its component functions are differentiable. Three important eamples of vector fields: A gradient field f (also called a conservative vector field) is a vector field that is the gradient of a multivariate function f (called the potential function of the field). We have already met gradient fields, when we studied derivatives of multivariate functions. Recall that: Gradient fields are orthogonal to the level sets of the potential function. At a point, the gradient points in the direction in which the rate of increase of the potential function is the greatest. The magnitude of the gradient at a point is the maimum rate of increase of the potential function at that point. 3

4 A radial field is a vector field that radiates from (or towards) the origin. Some eamples of radial fields are: r = î + y ĵ (outward-pointing radial field, -d) r = î y ĵ (inward-pointing radial field, -d) ˆr = + y î + y ĵ = cos θ î + sin θ ĵ + y (outward-pointing unit radial field, -d) r = î + y ĵ + z ˆk (outward-pointing radial field, 3-d) ˆr = + y î + y + y ĵ + (outward-pointing unit radial field, 3-d) z + y ˆk A rotational field (or spin field) is a vector field that rotates around the origin in -d. Some eamples of rotational fields are: Θ = y î + ĵ (counter-clockwise rotational field, -d) Θ = y î ĵ (clockwise rotational field, -d) ˆΘ = y + y î + ĵ = sin θ î + cos θ ĵ + y (counter-clockwise unit radial field, -d) A constant field is a vector field in which all vectors have the same direction and the same magnitude. In R, a constant field looks like F (, y) = a î + b ĵ, for some constants a and b. In R 3, a constant field looks like F (, y, z) = a î + b ĵ + c ˆk for a, b, and c constant. 4

5 Important note: Not every vector field is conservative, radial, rotational, or constant. These are just some important eamples of vector fields. 5

Worksheet 4.2: Introduction to Vector Fields and Line Integrals

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