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Chapter 1 Vector Calculus

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Chapter 1 Vector Calculus Chapter 1 vector calculus Vector calculus is the study of vector fields and related scalar functions. For the most part we focus our attention on two or three dimensions in this study. However, certain theorems are easily n extended to R . We explore these concepts in both Cartesian and the standard curvelinear coor- diante systems. I also discuss the dictionary between the notations popular in math and physics 1 The importance of vector calculus is nicely exhibited by the concept of a force field in mechanics. For example, the gravitational field is a field of vectors which fills space and somehow communi- cates the force of gravity between massive bodies. Or, in electrostatics, the electric field fills space and somehow communicates the influence between charges; like charges repel and unlike charges attract all through the mechanism propagated by the electric field. In magnetostatics constant magnetic fields fill space and communicate the magnetic force between various steady currents. All the examples above are in an important sense static. The source charges2 are fixed in space and they cause a motion for some test particle immersed in the field. This is an idealization. In truth, the influence of the test particle on the source particle cannot be neglected, but those sort of interactions are far too complicated to discuss in elementary courses. Often in applications the vector fields also have some time-dependence. The differential and inte- gral calculus of time-dependent vector fields is not much different than that of static fields. The main difference is that what was once a constant is now some function of time. A time-dependent vector field is an assignment of a vectors at each point at each time. For example, the electric and magnetic fields that together make light. Or, the velocity field of a moving liquid or gas. Many other examples abound. The calculus for vector fields involves new concepts of differentiation and new concepts of integra- tion. 1this version of my notes uses the inferior math conventions as to be consitent with earlier math courses etc... 2the concept of a charge really allows for electric, magnetic or gravitational although isolated charges exist only for two of the aformentioned 1 2 CHAPTER 1. VECTOR CALCULUS For differentiation, we study gradients, curls and divergence. The gradient takes a scalar field and generates a vector field (actually, this is not news for us). The curl takes a vector field and generates a new vector field which says how the given vector field curls about a point. The divergence takes a given vector field and creates a scalar function which quantifies how the given vector field diverges from a point. Many novel product rules exist for these operations and the algebra which links these operations is rich and interesting as we already saw at the beginning of this course as we studied vectors at a point. On the topic of integration there are two types of integration we naturally consider for a vector field. We can integrate along an oriented curve, this type of integration is called a line integral even when the curve is not a line. Also, for a given surface, we can calculate a surface integral of a vector field. The line integral measures how the vector field lines up along the curve of inte- gration, the measures something called the circulation. The surface integrals value depends on how the vector field pokes through the surface of integration, it measures something called flux. Critical to both linea and surface integrals are parametric equations for curves and surfaces. We'll see that we need parametrics to calculate anything yet the answers are completely indpendent of the parameters utilized. In other words, the surface and line integrals are coordinate free objects. This is in considerable contrast to the types of integrals we studied in the previous part of this course. Between the study of differentiation and integration of vector fields we find the unifying theorems of Greene, Gauss and Stokes. We study Green's theorem to begin since it is simple and merely a two- dimensional result. However, it is evidence of something deeper. Both Gauss and Stokes reduce to Greene's in certain context. There is also a fundmental theorem of line integrals which helps validate my claim that intuitively the rf is basically the derivative for a function of several variables. In addition, we will learn about how these integral theorems relate to the path-independence of vector fields. We saw earlier that certain vector fields could not be gradients because they violate Clairaut's Theorem on their components. On the other hand, just because a vector field has components consistent with Clairaut's theorem we also saw that it was not necessarily the case they were the gradient of some scalar function on their whole domain. We'll find sufficient conditions to make the Clairaut test work. We will find how to say with certainty a given vector field is the gradient of some scalar function. More than just that, we'll find a method to calculate the scalar function. 1.1. VECTOR FIELDS 3 1.1 vector fields Definition 1.1.1. n A vector field on S ⊆ R is an assignment of a n-dimensional vector to each point in S. If ~ Pn F = hF1;F2;:::;Fni = j=1 Fjxbj then the multivariate functions Fj : S ! R are called the component functions of F~ . In the cases of n = 2 and n = 3 we sometimes use the popular notations F~ = hP; Qi & F~ = hP; Q; Ri: We have already encountered examples of vector fields earlier in this course. ~ ~ 2 Example 1.1.2. Let F = xb or G = yb. These are constant vector fields on R . At each point we ~ ~ ~ ~ attach the same fixed vector; F (x; y) = h1; 0i or G(x; y) = h0; 1i. In contrast, H = rb and I = θb are 2 non-constant and technically are only defined on the punctured plane R − f(0; 0)g. In particular, x y −y x H~ (x; y) = ; & I~(x; y) = ; : px2 + y2 px2 + y2 px2 + y2 px2 + y2 2 Of course, given any differentiable f : S ⊆ R ! R we can create the vector field rf = h@xf; @yfi which is normal to the level curves of f. ~ ~ ~ 3 Example 1.1.3. Let F = xb or G = yb or H = zb. These are constant vector fields on R . At each point we attach the same fixed vector; F~ (x; y; z) = h1; 0; 0i or G~ (x; y; z) = h0; 1; 0i and ~ ~ ~ H(x; y; z) = h0; 1; 0i. In contrast, I = rb and J = θb are non-constant and technically are only 3 defined on R − f(0; 0; z) j z 2 Rg. In particular, x y −y x I~(x; y; z) = ; & J~(x; y; z) = ; : px2 + y2 px2 + y2 px2 + y2 px2 + y2 Similarly, the spherical coodinate frame ρ,b φ,b θb are vector fields on the domain of their definition. Of 3 course, given any differentiable f : S ⊆ R ! R we can create the vector field rf = h@xf; @yf; @zfi which is normal to the level surfaces of f. The flow-lines or streamlines are paths for which the velocity field matches a given vector field. Sometimes these paths which line up with the vector field are also called integral curves. Definition 1.1.4. In particular, given a vector field F~ = hP; Q; Ri we say ~r(t) = hx(t); y(t); z(t)i is an integral curve of F~ iff dx dy dz dx dx dz F~ (~r(t)) = ; ; a:k:a = P; = Q; = R: dt dt dt dt dt dt 4 CHAPTER 1. VECTOR CALCULUS We need to integrate each component of the vector field to find this curve. Of course, given that P; Q; R are typically functions of x; y; z the "integration" requires thought. Even in differential equations(334) the general problem of finding integral curves for vector fields is beyond our standard techniques for all but a handful of well-behaved vector fields. That said, the streamlines for the examples below are geometrically obvious so we can reasonably omit the integration. ~ Example 1.1.5. Suppose F = xb then obviously x(t) = xo + t; y = yo; z = zo is the streamline of xb through (xo; yo; zo). Likewise, I think you can calculate the steamlines for yb and zb without much trouble. In fact, any constant vector field F~ = ~vo simply has streamlines which are lines with direction vector ~vo. Example 1.1.6. The magnetic field around a long steady current in the postive z-direction is ~ µoI conveniently written as B(r; θ; z) = 2πr θb. The streamlines are circles which are centered on the axis and point in the θb direction. Example 1.1.7. If a charge Q is distributed uniformly through a sphere of radius R then the electric field can be show to me a function of the distance from the center of the sphere alone. Placing that center at the origin gives ( kρ 0 ≤ ρ ≤ R ~ R3 E(ρ) = ρb k ρ2 ρ ≥ R The streamlines are simply lines which flow radially out from the origin in all directions. Challenge: in electrostatics the density of streamlines (often called fieldlines in physics ) is used to measure the magnitude of the electric field. Why is that reasonable? Example 1.1.8. The other side of the thinking here is that given a differential equation we could use the plot of the vector field to indicate the flow of solutions.
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