Multivariable Calculus Math 21a Harvard University Spring 2004 Oliver Knill These are some class notes distributed in a multivariable calculus course tought in Spring 2004. This was a physics flavored section. Some of the pages were developed as complements to the text and lectures in the years 2000-2004. While some of the pages are proofread pretty well over the years, others were written just the night before class. The last lecture was "calculus beyond calculus". Glued with it after that are some notes from "last hours" from previous semesters. Oliver Knill. 5/7/2004 Lecture 1: VECTORS DOT PRODUCT O. Knill, Math21a VECTOR OPERATIONS: The ad- ~u + ~v = ~v + ~u commutativity dition and scalar multiplication of ~u + (~v + w~) = (~u + ~v) + w~ additive associativity HOMEWORK: Section 10.1: 42,60: Section 10.2: 4,16 vectors satisfy "obvious" properties. ~u + ~0 = ~0 + ~u = ~0 null vector There is no need to memorize them. r (s ~v) = (r s) ~v scalar associativity ∗ ∗ ∗ ∗ VECTORS. Two points P1 = (x1; y1), Q = P2 = (x2; y2) in the plane determine a vector ~v = x2 x1; y2 y1 . We write here for multiplication (r + s)~v = ~v(r + s) distributivity in scalar h − − i ∗ It points from P1 to P2 and we can write P1 + ~v = P2. with a scalar but usually, the multi- r(~v + w~) = r~v + rw~ distributivity in vector COORDINATES. Points P in space are in one to one correspondence to vectors pointing from 0 to P . The plication sign is left out. 1 ~v = ~v the one element numbers ~vi in a vector ~v = (v1; v2) are also called components or of the vector. ∗ REMARKS: vectors can be drawn everywhere in the plane. If a vector starts at 0, then the vector ~v = v ; v h 1 2i LENGTH. The length ~v of ~v is the distance from the beginning to the end of the vector. points to the point v1; v2 . That's is why one can identify points P = (a; b) with vectors ~v = a; b . Two j j vectors which can beh translatedi into each other are considered equal. In three dimensions, vectorsh havie three EXAMPLES. 1) If ~v = (3; 4) , then ~v = p25 = 5. 2) ~i = ~j = ~k = 1, ~0 = 0. components. In some Encyclopedias like Encylopedia Britannica define vectors as objects which have "both j j j j j j j j j magnitude and direction". This is unprecise and strictly speaking incorrect because the zero vector is also a UNIT VECTOR. A vector of length 1 is called a unit vector. If ~v = ~0, then ~v= ~v is a unit vector. vector but has no direction. 6 j j EXAMPLE: If ~v = (3; 4), then ~v = (2=5; 3=5) is a unit vector, ~i;~j;~k are unit vectors. ADDITION SUBTRACTION, SCALAR MULTIPLICATION. PARALLEL VECTORS. Two vectors ~v and w~ are called parallel, if ~v = rw~ with some constant r. y y y 3 u u u DOT PRODUCT. The dot product of two vectors ~v = (v ; v ; v ) and w~ = (w ; w ; w ) is defined as u+v u-v 1 2 3 1 2 3 u x x v v x ~v w~ = v w + v w + v w · 1 1 2 2 3 3 i Remark: in science, other notations are used: ~v · w~ = (~v; w~) (mathematics) < ~vjw~ > (quantum mechanics) viw (Einstein i j notation) gij v w (general relativity). The dot product is also called scalar product, or inner product. ~u + ~v = u ; u + v ; v ~u ~v = u ; u v ; v λ~u = λ u ; u h 1 2i h 1 2i − h 1 2i − h 1 2i h 1 2i = u1 + v1; u2 + v2 = u1 v1; u2 v2 = λu1; λu2 LENGTH. Using the dot product one can express the length of ~v as ~v = p~v ~v. h i h − − i h i j j · CHALLENGE. Express the dot product in terms of the length alone. BASIS VECTORS. The vectors ~i = 1; 0 , ~j = 0; 1 are called standard basis vectors in the plane. In space, h i h i one has the basis vectors ~i = 1; 0; 0 , ~j = 0; 1; 0 , ~k = 0; 0; 1 . SOLUTION: (~v + w~; ~v + w~) = (~v; ~v) + (w~; w~) + 2(~v; w~) can be solved for (~v; w~). h i h i h i Every vector ~v = (v1; v2) in the plane can be written as a sum of standard basis vectors: ~v = v1~i + v2~j. Every vector ~v = (v1; v2; v3) in space can be written as ~v = v1~i + v2~j + v3~k. 2 2 2 WHERE DO VECTORS OCCUR? Here are some examples: ANGLE. Because ~v w~ = (~v w~; ~v w~) = ~v + w~ 2(~v; w~) is by the cos-theorem equalj −to j~v 2 + w~−2 2−~v w~ jcosj (α),j wherej − α is the angle between the vectors ~v andj j w~, jwej get− thej j ·impj jortant formula Forces: Some prob- Velocity: if (f(t); g(t)) is a point in the plane lems in statics in- ~v w~ = ~v w~ cos(α) volve the determina- · j j · j j which depends on time t, 0 0 tion of a forces acting then ~v = f (t); g (t) is the velocith y vectori at on objects. Forces are represented as CAUCHY-SCHWARZ INEQUALITY: ~v w~ ~v w~ follows from that formula because cos(α) 1. the point (f(t); g(t)). vectors TRIANGLE INEQUALITY: ~u + ~v j~u· +j ~v≤ jfollojj wsj from ~u + ~v 2 = (~u + ~v) (~u + ~v) j= ~u2 +j ~v≤2 + 2~u ~v ~u2 + ~v2 + 2 ~u ~v ~u2 + ~v2 +j2 ~u j~v≤=j (j~u +j j~v )2. j j · · ≤ j · j ≤ j j · j j j j j j Fields: fields like elec- Qbits: in quantum FINDING ANGLES BETWEEN VECTORS. Find the angle between the vectors (1; 4; 3) and ( 1; 2; 3). tromagnetic or gravita- computation, one ANSWER: cos(α) = 16=(p26p14) 0:839. So that α = arccos(0:839::) 33◦. − tional fields or velocity does not work with ∼ ∼ fields in fluids are de- bits, but with qbits, orthogonal ~ scribed with vectors. which are vectors. ORTHOGONAL VECTORS. Two vectors are called if v w = 0. The zero vector 0 is orthogonal to any vector. EXAMPLE: ~v = (2; 3) is orthogonal to w~ = ( 3; 2). · − Color can be written as w 2 a vector ~v = (r; g; b), blue SVG. Scalable PROJECTION. The vector ~a = projw~ (~v) = w~(~v w~= w~ ) is called the · j j b where r is red, g is green (r,g,b) Vector Graphics is projection of ~v onto w~. and b is blue. An other an emerging stan- The scalar projection is defined as compw~(~v) = (~v w~)= w~ . (Its ab- coordinate system is ~v = dard for the web · j j green solute value is the length of the projection of ~v onto w~.) The vector (c; m; y) = (1 r; 1 for describing two- − − red ~ component g; 1 b), where c is cyan, dimensional graphics b = ~v ~a is called the of ~v orthogonal to the w~-direction. v − − a m is magenta and y is in XML. yellow. EXAMPLE. ~v = (0; 1; 1), w~ = (1; 1; 0), projw~ (~v) = (1=2; 1=2; 0), compw~ (~v) = 1=p2. − − − Lecture 2: CROSS PRODUCT Math21a, O. Knill HOMEWORK: section 10.2: 28,42, section 10.3: 12,20 DISTANCE POINT-LINE (3D). If P is a point in space and L is the line which contains the vector ~u, then CROSS PRODUCT. The cross product of two vectors d(P; L) = jP~Q × ~uj=j~uj ~v = hv1; v2; v3i and w~ = hw1; w2; w3i is defined as the vector ~v × w~ = hv2w3 − v3w2; v3w1 − v1w3; v1w2 − v2w1i. is the distance between P and the line L. To compute it: v1 v2 v3 v1 v2 multiply diagonally X X X PLANE THROUGH 3 POINTS P; Q; R: at the crosses. w1 w2 w3 w1 w2 The vector ~n = P~Q × P~R is orthogonal to the plane. We will see next week that ~n = (a; b; c) defines the plane ax + by + cz = d, with d = ax0 + by0 + cz0 which passes through the points P = (x0; y0; z0); Q; R. DIRECTION OF ~v × w~: ~v × w~ is orthogonal to ~v and orthogonal to w~. The cross product appears in many different applications: ANGULAR MOMENTUM. If a mass point of mass m moves along a curve ~r(t), then the vector L~ (t) = Proof. Check that ~v · (~v × w~) = 0. 0 m~r(t) × ~r (t) is called the angular momentum of the point. It is coordinate system dependent. LENGTH: j~v × w~j = j~vjjw~j sin(α) ANGULAR MOMENTUM CONSERVATION. 2 2 2 2 Proof. The identity j~v×w~j = j~vj jw~j −(~v·w~) can be proven by direct computation. Now, j~v·w~j = j~vjjw~j cos(α). d 0 0 00 L~ (t) = m~r (t) × ~r (t) + m~r(t) × ~r (t) = ~r(t) × F~ (t) dt AREA. The length j~v × w~j is the area of the parallelogram spanned by ~v and w~. In a central field, where F~(t) is parallel to ~r(t), we get d=dtL(t) = 0 which means L(t) is constant. Proof. Because jw~j sin(α) is the height of the parallelogram with base length j~vj, the area is j~vjjw~j sin(α) which 0 is by the above formula equal to j~v × w~j. TORQUE.