18.02 Review Jeremy Orloff. 1 Review of multivariable calculus (18.02) constructs

Transcription

1 18.02 eview Jerem Orloff 1 eview of multivariable calculus (18.02) constructs 1.1 Introduction These notes are a terse summar of what we ll need from multivariable calculus. If, after reading these, some parts are still unclear, ou should consult our notes or book from our multivariable calculus or ask about it at office hours. We ve also posted a more detailed review of line integrals and Green s theorem. You should consult that if needed. We ve seen that comple eponentials make trigonometric functions easier to work with and give insight into man of the properties of trig functions. Similarl, we ll eventuall reformulate some material from in comple form. We ll see that it s easier to present and the main properties are more transparent in comple form. 1.2 Terminolog and notation Vectors. We ll denote vectors in the plane b (, ) Note. In phsics and in we usuall write vectors in the plane as i + j. This use of i and j would be confusing in 18.04, so we will write this vector as (, ). In ou might have used angled brackets, for vectors and round brackets (, ) for points. In we will adopt the more standard mathematical convention and use round brackets for both vectors and points. It shouldn t lead to an confusion. Orthogonal. Orthogonal is a snonm for perpendicular. Two vectors are orthogonal if their dot product is zero, i.e. v = (v 1, v 2 ) and w = (w 1, w 2 ) are orthogonal if v w = (v 1, v 2 ) (w 1, w 2 ) = v 1 w 1 + v 2 w 2 = 0. omposition. omposition of functions will be denoted f(g(z)) or f g(z), which is read as f composed with g 1.3 Parametrized curves We often use the greek letter gamma for a paramtrized curve, i.e. γ(t) = ((t), (t)). We think of this as a moving point tracing out a curve in the plane. The tangent vector γ (t) = ( (t), (t)) is tangent to the curve at the point ((t), (t)). It s length γ (t) is the instantaneous speed of the moving point. 1

2 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS 2 γ (t) γ(t) γ (t) Parametrized curve γ(t) with some tangent vectors γ (t). Eample ev.1. Parametrize the straight line from the point ( 0, 0 ) to ( 1, 1 ). answer: There are alwas man parametrizations of a given curve. A standard one for straight lines is γ(t) = (, ) = ( 0, 0 ) + t( 1 0, 1 0 ), with 0 t 1. Eample ev.2. Parametrize the circle of radius r around the point ( 0, 0 ). answer: Again there are man parametrizations. Here is the standard one with the circle traversed in the counterclockwise direction: γ(t) = (, ) = ( 0, 0 ) + r(cos(t), sin(t)), with 0 t 2π. r ( 0, 0 ) ( 1, 1 ) Line from ( 0, 0 ) to ( 1, 1 ) and circle around ( 0, 0 ). 1.4 hain rule For a function f(, ) and a curve γ(t) = ((t), (t)) the chain rule gives df(γ(t)) = f dt (t) + f γ(t) (t) = f(γ(t)) γ (t) dot product of vectors. γ(t) Here f is the gradient of f defined in the net section. 1.5 Grad, curl and div Gradient. For a function f(, ), the gradient is defined as gradf = f = (f, f ). A vector field F which is the gradient of some function is called a gradient vector field.

3 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS 3 url. For a vector in the plane F(, ) = (M(, ), N(, )) we define curlf = N M. Note. The curl is a scalar. In and in general, the curl of a vector field is another vector field. However, for vectors fields in the plane the curl is alwas in the k direction, so we have simpl dropped the k and made curl a scalar. Divergence. The divergence of the vector field F = (M, N) is divf = M + N. 1.6 Level curves ecall that the level curves of a function f(, ) are the curves given b f(, ) = constant. ecall also that the gradient f is orthogonal to the level curves of f 1.7 Line integrals The ingredients for line (also called path or contour) integrals are the following: A vector field F = (M, N) A curve γ(t) = ((t), (t)) defined for a t b Then the line integral of F along γ is defined b γ b a F(γ(t)) γ (t)dt = γ Md + Nd. Eample ev.3. Let F = ( /r 2, /r 2 ) and let γ be the unit circle. ompute line integral of F along γ. answer: You should be able to suppl the answer to this eample Properties of line integrals 1. Independent of parametrization. 2. everse direction on curve change sign. That is, F dr. (Here, means the same curve traversed in the opposite direction.) 3. If is closed then we sometimes indicate this with the notation M d + N d.

4 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS Fundamental theorem for gradient fields Theorem ev.4. (Fundamental theorem for gradient fields) If F = f then γ f(p ) f(q), where Q, P are the beginning and endpoints respectivel of γ. Proof. B the chain rule we have df(γ(t)) dt = f(γ(t)) γ (t) = F(γ(t)) γ (t). The last equalit follows from our assumption that F = f. Now we can this when we compute the line integral: γ = b a b a F(γ(t)) γ (t) dt df(γ(t)) dt dt = f(γ(b)) f(γ(a)) = f(p ) f(q) Notice that the third equalit follows from the fundamental theorem of calculus. Definition. If a vector field F is a gradient field, with F = f, then we call f a a potential function for F. Note: the usual phsics terminolog would be to call f the potential function for F Path independence and conservative functions Definition. For a vector field F, the line integral F dr is called path independent if, for an two points P and Q, the line integral has the same value for ever path between P and Q. Theorem. F dr is path independent is equivalent to 0 for an closed path. Sketch of proof. Draw two paths from Q to P. Following one from Q to P and the reverse of the other back to P is a closed path. The equivalence follows easil. We refer ou to the more detailed review of line integrals and Green s theorem for more details. Definition. A vector field with path independent line integrals, equivalentl a field whose line integrals around an closed loop is 0 is called a conservative vector field. Theorem ev.5. We have the following equivalence: On a connected region, a gradient field is conservative and a conservative field is a gradient field. Proof. Again we refer ou to the more detailed review for details. Essentiall, if F is conservative then we can define a potential function f(, ) as the line integral of F from some base point to (, ).

5 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS Green s Theorem Ingredients: a simple closed curve (i.e. no self-intersection), and the interior of. must be positivel oriented (traversed so interior region is on the left) and piecewise smooth (a few corners are oka). Theorem ev.6. Green s Theorem: If the vector field F = (M, N) is defined and differentiable on then M d + N d = N M da. In vector form this is written where the curl is defined as curlf = (N M ). curlf da. Proof of Green s Theorem. See the more detailed notes on Green s theorem and line integrals for the proof. 1.9 Etensions and applications of Green s theorem Simpl connected regions Definition: A region D in the plane is simpl connected if it has no holes. Said differentl, it is simpl connected for ever simple closed curve in D, the interior of is full contained in D. Eamples: D 1 D 2 D 3 D 4 D 5 = whole plane D1-D5 are simpl connected. For an simple closed curve inside an of these regions the interior of is entirel inside the region.

6 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS 6 Note. Sometimes we sa an curve can be shrunk to a point without leaving the region. The regions below are not simpl connected. For each, the interior of the curve is not entirel in the region. Annulus Punctured plane Potential Theorem Here is an application of Green s theorem which tells us how to spot a conservative field on a simpl connected region. The theorem does not have a standard name, so we choose to call it the Potential Theorem. Theorem ev.7. (Potential Theorem) Take F = (M, N) defined and differentiable on a region D. (a) If F = f then curlf = N M = 0. (b) If D is simpl connected and curlf = 0 on D, then F = f for some f. We know that on a connected region, being a gradient field is equivalent to being conservative. So we can restate the Potential Theorem as: on a simpl connected region, F is conservative is equivalent to curlf = 0. Proof of (a): F = (f, f ), so curlf = f f = 0. Proof of (b): Suppose is a simple closed curve in D. Since D is simpl connected the interior of is also in D. Therefore, using Green s theorem we have, curlf da = 0. D This shows that F is conservative in D. Therefore, b Theorem ev.5 F is a gradient field. Summar: Suppose the vector field F = (M, N) is defined on a simpl connected region D. Then, the following statements are equivalent. (1) (2) Q P F dr is path independent. 0 for an closed path.

7 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS 7 (3) F = f for some f in D (4) F is conservative in D. If F is continuousl differentiable then 1,2,3,4 all impl 5: (5) curlf = N M = 0 in D Wh we need simpl connected in the Potential Theorem If there is a hole then F might not be defined on the interior of. (See the eample on the tangential field below.) D Etended Green s Theorem We can etend Green s theorem to a region which has multiple boundar curves. Suppose is the region between the two simple closed curves 1 and (Note is alwas to the left as ou traverse either curve in the direction indicated.) Then we can etend Green s theorem to this setting b F dr + curlf da. 1 2 Likewise for more than two curves: F dr + F dr + F dr curlf da. Proof. The proof is based on the following figure. We cut both 1 and 2 and connect them b two copies of 3, one in each direction. (In the figure we have drawn the two copies of 3 as separate curves, in realit the are the same curve traversed in opposite directions.) Now the curve = is a simple closed curve and Green s theorem holds on it. But the region inside is eactl and the contributions of the two copies of 3 cancel. That is, we have shown that curlf da = F dr

8 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS 8 This is eactl Green s theorem, which we wanted to prove (, ) Eample ev.8. Let F = r 2 The punctured plane. ( tangential field ) F is defined on D = plane - (0,0) = the punctured plane. (Shown below.) It s eas to compute (we ve done it before) that curlf = 0 in D. Question: For the tangential field F what values can F dr take for a simple closed curve (positivel oriented)? answer: We have two cases (i) 1 not around 0 (ii) 2 around In case (i) Green s theorem applies because the interior does not contain the problem point at the origin. Thus, curlf da = 0. 1 For case (ii) we will show that 2π. 2 Let 3 be a small circle of radius a, entirel inside 2. B the etended Green s theorem we have F dr curlf da =

9 1 EVIEW OF MULTIVAIABLE ALULUS (18.02) ONSTUTS 9 Thus, 2 F dr. 3 Using the usual parametrization of a circle we can easil compute that the line integral is 3 2π 0 1 dt = 2π. QED. 3 2 Answer to the question: The onl possible values are 0 and 2π. We can etend this answer in the following wa: If is not simple, then the possible values of F dr are 2πn, times goes (counterclockwise) around (0,0). where n is the number of Not for class: n is called the winding number of around 0. n also equals the number of times crosses the positive -ais, counting +1 from below and 1 from above.