Some Important Concepts and Theorems of Vector Calculus

Transcription

1 ome Important oncepts and Theorems of Vector alculus 1. The Directional Derivative. If f(x) is a scalar-valued function of, say x, y, and z, andu is a direction (i.e. a unit vector), then the Directional derivative of f at the point P, inthedirectionu is f(p + hu) f(p ) D u f(p )= lim. h 0 h When u is the basic vector i, (orj or k), then the directional derivative becomes the partial derivative f (or f or f ). P z P 2. The Gradient. If f(x) is a scalar-valued function, say of three variables x, y, z, andp is a point, then grad P (f) is a vector whose direction maximizes the directional derivative at P, and whose length is the value of the directional derivative in the maximizing direction. ( grad f points in the direction of maximal increase of f ); grad f is also denoted f and we have the important formulas: f grad P f =, f, f À = f i+ f j+ f k D u f(p ) = (grad P f) u (Directional derivative is grad dot direction). 3. The Line Integral. Let be a curve with the parametrization r(t) =hx(t),y(t),z(t)i, witha t b. Let F = hp (x, y, z), Q(x, y, z), (x, y, z)i = P i+qj+k be a flow or force field. The line integral of F along takes small pieces of arc ds, multiplies the component of F in the direction of this arc times the length of the arc, and adds up all these pieces along the entire curve. IfwedenotebyF(t) what we get by putting x = x(t),y = y(t),z = z(t) into F(x, y, z), then we have the formula b F ds = F(t) hx 0 (t),y 0 (t),z 0 (t)i dt. a When F is a flow field, this is the circulation along ; whenf is a force field, this is the work along. 4. Path Independent Fields. Avectorfield F is said to be path independent in some region of space if, for any points P and Q in, andany curve beginning at P and ending at Q, you always get the same value for F ds. 5. onservative Fields. Avectorfield F is called conservative in some region if, for any closed curve (i.e. beginning and ending at the same point) in, I F ds =0. 6. Gradient or Potential Fields. F is a gradient fieldinaregionif there is a scalar-valued function f such that, for every point P in, F(P ) = grad P f (= f(p )). If F = P i + Q j + k this means that P = f x, Q = f f,and = z. 1

2 7. Fundamental Equivalence. uppose that F is continuous in a region in which any two points can be joined by a smooth curve. Then the properties of Path Independence, being onservative, and being a Gradient Field, are all equivalent (i.e. if one of these properties holds, they all hold, and if one of these properties is false, they are all false). 8. Fundamental Theorem for Line Integrals. If F =gradf, and is a curve beginning at P 0 and ending at P 1,then F ds = grad f ds = f(p 1 ) f(p 0 ). 9. Green s Theorem. (This holds only in the plane, i.e. for two variables.) Let F(x, y) =P (x, y) i+q(x, y) j be a vector field and let be a bounded region which has no holes and in which P and Q are continuous and have continuous partial derivatives. Let be a closed curve in parametrized so that A, the region it bounds, is always to the left as r(t) moves along the curve. Then I I µ Q F ds = Pdx+ Qdy= x P da. 10. Important onsequence of Green s Theorem. If is a region in which Green s Theorem applies, then F is a gradient field if, and only if, Q x P Q =0. The scalar function x P is called the two-dimensional curl of the vector field F. Thus,F =gradf if, and only if Q x = P. 11. urface Integrals. Let be a surface in space and F a vector field, interpreted as the velocity of a flow or simply called a flow (measured, say, in meters per second). We suppose the surface comes with vector field n which is continuous, length 1, and normal or perpendicular to the surface at every point. uch a field is often called an orientation or an outward pointing normal. The surface integral takes a small piece of surface area d, takes the (scalar) component of F in the outward normal direction (i.e takes. F n), multiplies this by the area d, and adds all these quantities together for all d on the surface. Each of these these piece (F n) d is the flux of F through d, and its units are, say, (meters per second)(square meters) or volume per unit time. The surface integral is denoted F n d, and is called the flux of F through. 12. omputing urface Integrals. We take a surface with a parametrization hx(s, t), y(s, t), z(s, t)i (so x = x(s, t), y = y(s, t), andz = z(s, t)) wheres and t take on values in the region A in 2-space. x Let v 1 = s, s, z À x and v 2 = s t, t, z À.upposeF = P i + Q j + j, and denote by F(s, t) t the result of substituting for x, y, z their values in terms of s and t. Then F n d = F(s, t) (v 1 v 2 ) da = A P (s, t) Q(s, t) (s, t) x z det s s s x z A t t t A da 2

3 13. The Divergence. Let F(x, y, z) be a vector field and P a point in space. Take a small ball of radius h around P,computetheflux of F outward through the spherical surface bounding the ball use a surface integral to do this and divide by the volume of the ball. This is the average flux per unit volume near P, or the average flux density. Now take the limit of this average flux density as the radius h goes to 0. This is the flux density at P, also called the divergence at P. It is a measure of the amount of fluid flowing away from the point P, and is denoted div P F or ( F)(P ) 14. omputing the Divergence. If F = P i + Q j + j, then µ P div P0 F == x + Q + z P The Divergence Theorem. uppose V is a bounded volume in space; for example, a ball, or a solid ellipsoid, or a simple blob. uppose is the surface of V, oriented so that the normal points outward or away from V. LetF be a flow field whose components P, Q, and have continuous partial derivatives throughout V. Then F n d = (div F) dv. This says that the flux of F through the surface is equal to the total divergence of F throughout the interior. Or, put another way, whatever flows through outward through the surface (in unit time) must have originated at some point in the interior. This is a kind of formal, mathematical, conservation law. 16. Why is the Divergence Theorem True? emember that the divergence at a point is obtained by taking a small volume (ball) around the point, dividing the flux through the surface of the ball by its volume, and taking the limit as the volume squeezes down to the point. oughly speaking, then, if dv is a small volume bounded by the small surface d, then (div F) dv = µ flux through d volume dv V (volume dv) =flux through d. Thus, to compute (div F) dv, we can cut up ( dice ) the volume V into little cube-like regions V µ flux through d dv and add up all the pieces: (volume dv). Now, if two of these little cube-like volume dv regions share a common face, the flux out of one of these faces is exactly equal to the flux into the adjacent face. One flux will be counted positive, the other negative since one is in, the other out so they will cancel out. Thus, the flux through the surface made up of the surfaces of both of these cube-like regions is simply the flux through the outer faces: the flux through the inner face amounting to 0. Asweaddinmoreandmoreofthecube-likeregions,weseethatthesumofthefluxes through their surfaces will always amount to just the flux through their (net) boundary surfaces, with all the fluxes through interior surfaces cancelling. Thus, when we take the total sum, we will be left with just the flux through the total boundary, namely (approximately) the flux through the boundary surface. The sum, then, that approximates (div F) dv is seen also to approximate F n d, and so that is why these integrals are equal. V 3

4 17. The Electric Field. The electric fieldduetoachargeq (in coulombs) at the origin is given by: µ 1 q r E(x, y, z) =E(r) = 4π r 3. where is a constant called the permittivity of free space. uppose is a sphere of radius around the origin. Then E is normal (perpendicular) to at every point on, sincee points basically in the direction of r. Furthermore, the magnitude of E on is given by E(r) = 1 q r 4π r 3 = 1 r q 4π r 2 = 1 q 4π 2.Thus Flux through the sphere = E n d = 1 q 4π 2 d = 1 q 4π 2 1 d = 1 4π = q. q 2 (Area of )= 1 4π q 4π 2 2 Now it is easy enough to compute, explicitly from the formula for divergence, that div E =0. But this seems to violate the Divergence Theorem; we should have E n d = (div E) dv =0. V However, this is not a contradiction since div E =0only for points not equal to the origin i.e., the place where the discrete charge lies! Thefield E is not even defined at (0, 0, 0), so the Divergence Theorem does not apply to a sphere containing a point charge at the origin. What is remarkable, is that we can get a useful result from the Divergence Theorem anyway. uppose we have any surface which is the boundary of a nice region V containing the origin (so the origin is inside ). Now take to be a sphere centered at the origin, and so big that it contains with room to spare. Let be the solid region between and.. Then the boundary of is made of two pieces: an outer surface andaninnersurface. (For example, if is a sphere of radius 2 and = 3 = asphereofradius3, then is just the solid shell between these two spheres.) Let us now orient so that its normal vector points away from ; thus, its normal vector must point towards the origin. ince this orientation is the opposite of the one which points outward from, wecall with this orientation. Let s orient so that its normal points away from also, so its normal vector point away from the origin: the usual orientation for a sphere. We can now apply the Divergence Theorem to the region (which doesn t contain the origin) and its oriented boundary made up of the two surface and : E n d = (div E) dv. The left-hand integral is simply E n d E n d, while the right-hand one is simply 0 because doesn t contain the origin, so div E =0everywhere in. Thusweseethat E n d = E n d = q. 4

5 This proves a special case of Gauss Law which states that the electric flux through any surface containing total charge q is always q. Bywhatwehaveproved,thisistrueforfinite sums of points charges. However, when charge is distributed by huge numbers of tiny point charges (e.g. electrons) we often model the situation by using a continuous charge distribution. To do this, we first define charge in volume V ρ(x, y, z), thecharge density at (x, y, z), tobelim V 0, where the limit is taken V of small volumes containing (x, y, z). Then the total charge within the region is 1 ρdv.the electric field for such a distribution of charge is given by: E(r) = 1 ρ(x)x 4π r x 3 dv all space This field E is actually continuous for reasonable continuous distributions of charge (i.e. for reasonable ρ). Thus, the Divergence Theorem does apply here, and we conclude that E n d = (div E) dv. On the other hand, because 1 says: ρdv is the total charge in, Gauss Law, in its general form, E n d = 1 ρdv. ince (div E) dv = 1 ρdv all simple volumes and their bounding surfaces,we deduce Gauss Law in its differential form (one of Maxwell s equations): div E = ρ 18. The Magnetic Field. The situation here is quite a bit different and somewhat more complicated. For electricity, there exist isolated charges; for example, electrons. However, there does not seem to be any such thing for magnetism. Magnets come in paired pieces: a north and a south pole. uch a pair is called a dipole, and the magnetic field B flows from one pole to the other in closed arcs, which can actually be visualized by sprinkling fine iron filings over a piece of paper covering a magnet. As a result, one can write a formula for the magnetic field for such a dipole, and show that magnetic flux through any sphere is simply 0. onsequently,theflux density is 0 as well, and we get the fundamental law (also one of Maxwell s equations): div B = The url of a Vector Field. If is I a closed curve in space, and F is a vector field, then the circulation of F around is given by F ds. Now let P be a point in space and let V be any 5

6 vector. Let V P be the plane perpendicular to V and passing through P. Let be a small circle of radius h lying in V P with P as its center. The average circulation density of F at P around V is defined to be the circulation around divided by the area enclosed by. Note that this just depends on the direction of V, not its magnitude. Now take the limit of this as h 0; theirculation Density of V at P, in the direction V is this limit. The url of F at P is the vector V whose direction maximizes this circulation density, and whose magnitude is equal to this maximal circulation density. (ompare this with the definition of gradient given above.) o curl P F is a vector that points in the direction of maximum circulation density, and whose magnitude is that circulation density. 20. omputing the url. uppose F =P i+q j+ k and P 0 is a point in space. Then curl F= F =det where all these quantities are evaluated at P 0. It is also true that = i j k x z P Q µ Q µ P i+ z z µ Q j+ x x P k irculation Density at P in the direction V is (curl P F) V. (This is similar to computing the directional derivative by dotting with the gradient.) 21. tokes Theorem. Let be a finite (bounded) piece of surface bounded by the smooth curve. We suppose that has an orientation n. uppose is parametrized in such a way that as one walks around the curve with one s head pointing in the direction n, one s left hand is in the surface. Then I F ds = (curl F) n d. This says that the circulation around a closed curve is equal to the integral of the curl over the surface it bounds. Of course, we have to assume, as usual, that the components P, Q, of F have continuous partials. 22. Why is tokes Theorem True? Let s look at the surface integral (curl F) n d. Thequantity (curl F) n is simply the circulation density in the direction n, so, for a small closed curve d enclosing circulation around d area d on the surface, it is roughly:, so this surface integral is simply adding µ area d circulation around d up quantities d = circulation around d. area d Now think of the surface as being divided up into small curved square-like regions (say by dividing up the parameter region into small squares), each with area d and with bounding curve d. Iftwosuch square-like regions are adjacent, they share an edge. If they are both oriented say counterclockwise, then their orientations of the shared edge are opposite, so the flow or circulations on this edge cancel 6

7 each other out. Thus, the circulation around a region formed by joining two adjacent regions is just the circulation around the outer boundary, with the circulation around the interior edge cancelling out. As we add more and more of these curved regions to our sum, we see that the result is always the circulation around the outermost edges, with the circulations around the inner edges always cancelling. Thus, taking the aggregate of all these these regions i.e. approximating the integral over the whole surface I we end up with just the circulation around the outermost boundary, which is roughly F ds. ince these sums approximate both integrals, the integrals are equal in the limit. This is why tokes Theorem is true. ompare this argument with the argument given above for the Divergence Theorem. 23. tokes Theorem and Green s Theorem. uppose that the surface in tokes µ Theorem lies entirely Q in the xy-plane,asdoesthevectorfield F (i.e. F = P i + Q j). Then curl F= x P k, andthe surface integral becomes an ordinary double integral, so we get Green s Theorem in the plane. (In fact, it is possible to give a mathematically rigorous proof of tokes Theorem using just Green s theorem.) 24. onsequences of tokes Theorem. We firstnotethatifcurl F =0, and the partials of F are all continuous in a regions, then the circulation of F around any closed curve in is 0. Therefore F is agradientfield. o, for regions such as, curl F =0implies F =gradf for some f. This generalizes a result we stated above, obtained using Green s Theorem. It turns out that the converse is also true: if F =gradf then curl F =0;thisisbecauseitis possible to prove (using the equality of mixed partials also known as lairaut s Theorem) that curl(grad F)) = 0. (It is also a fact that div(curl F)) = 0). uppose now that 1 and 2 are two curves that don t meet, and together form the boundary of asurface on which curl F =0. We suppose that their orientations are such that when we walk around them so that our head points in the normal direction for, the region is on our left. Then F ds = F ds, since the sum of these integrals is equal to the integral of the curl, 1 2 whichwehaveassumedtobe0. 7