Math S21a: Multivariable calculus Oliver Knill, Summer 20111: Geometry and DistanceThe arena for multivariable calculus is the two-dimensional plane and the three dimensionalspace.A point in the plane has two coordinates P = (x, y). A point in space is de-termined by three coordinates P = (x, y, z). The signs of the coordinates define 4quadrants in the plane and 8 octants in space. These regions by intersect at theorigin O = (0, 0) or O = (0, 0, 0) and are separated by coordinate axes {y = 0 }and {x = 0 } or coordinate planes {x = 0 }, {y = 0 }, {z = 0 }.In two dimensions, the x-coordinate usually directs t o the ”east” and the y-coordinate points”north”. In three dimensions the usual coordinate system has the xy-plane as the ”ground” andthe z-coordinate axes pointing ”up”.1 P = (2, −3) is in the forth quadrant of the plane and P = (1, 2, 3) is in the positive octantof space. The point (0, 0, −5) is on the negative z axis. The point (1, 2, −3) is below thexy-plane.2 Pr oblem. Find the midp oint M of P = (1, 2, 5) a nd Q = (−3, 4, 7). Answer. The midpointis obtained by taking the average of each coordinate M = (P + Q)/2 = (−1, 3, 6).3 In computer graphics of photography, the xy-plane cont ains the retina or film plate. The zcoordinate measures the distance towa r ds the viewer. In this photographic coordinatesystem your eyes and mouth are in t he plane z = 0 and your nose points in the z direction.If the midpoint of your eyes is the origin of the coordinate system and your eyes havethe coo rdinates (1, 0, 0), (−1, 0, 0), then the tip of your nose might have the co ordinates(0, −1, 1).The Euclidean distance between two points P = (x, y, z) and Q = (a, b, c) inspace is defined as d(P, Q) =q(x − a)2+ (y − b)2+ (z − c)2.This Euclidean distance is a definition but motivated by Pythagoras theorem.4 Pr oblem: Find the distance d(P, Q) between the points P = (1 , 2, 5) and Q = (−3, 4, 7)and verify that d(P, M) + d(Q, M) = d(P, Q). Answer: The distance is d(P, Q) =√42+ 22+ 22=√24. The distance d(P, M) is√22+ 12+ 12=√6. The distance d(Q, M)is√22+ 12+ 12=√6. Indeed d(P, M) + d(M, Q) = d(P, Q).Remarks.1) Distances can be introduced more abstractly: take any nonnegative function d(P, Q) whichsatisfies the triangle inequality d(P , Q) + d(Q, R) ≥ d(P, R) and d(P, Q) = 0 if and only ifP = Q. A set X with such a distance function d is called a metric space. Examples ofdistances are the Manhatten distance dm(P, Q) = |x − a| + |y − b|, the quartic distanced4(P, Q) = ((x − a)4+ (y − b)4or the Fermat distance df(x, y) = d(x, y) if y > 0 anddf(x, y) = 1.33d(x, y) if y < 0. The constant 1.33 is the refractive index and models theupper half plane being filled with air a nd the lower half plane with water. Shortest paths are bentat the water surface. Each of these distances d, dm, d4, dfmake the plane a different metric space.2) It is symmetry which distinguishes the Euclidean distance as the most natural one. The Eu-clidean distance is determined by d((1, 0, 0), (0, 0, 0) ) = 1, rotational and translational and scalesymmetry d(λP, λQ) = λd(P, Q).3) We usually work with a right handed coordinate system, where the x, y, z axes can bematched with the thumb, po inting and middle finger of the right hand. The photographerscoordinate system is an example of a left handed coordinate system. The x, y, z axes arematched with the thumb and pointing finger and middle fing er of the left hand. Nature is notoblivious to parity. Some laws of particle physics are different when they are observed in a mirror.Coordinate systems with different parity can not be rotated into each other.Points, curves, surfaces and solid bodies are geometric objects which can be described withfunctions of several variables. An example of a curve is a line, an example of a surface is aplane, an example of a solid is the interior of a sphere. We focus in this first lecture on spheres orcircles.A circle of radius r centered at P = (a, b) is the collection of points in the pla newhich have distance r from P .A sphere of radius ρ centered at P = (a, b, c) is the collection of points in spacewhich have distance ρ from P . The equation of a sphere is (x−a)2+(y−b)2+(z−c)2=ρ2.An ellipse is the collection of points P in the plane for which the sum d(P, A) +d(P, B) of the distances to two points A, B is a fixed constant l larger than d(A, B).This allows to draw the ellipse with a string of length l attached at A, B. Analgebraic equivalent description is the set of p oints satisfying an equation x2/a2+y2/b2= 1.5 Pr oblem: Is the point (3, 4, 5) outside or inside the sphere (x−2)2+(y −6)2+(z −2)2= 16?Answer: The distance of the point to the center of the sphere is√1 + 4 + 9 which is smallerthan 4 the radius of the sphere. The point is inside.6 Pr oblem: Find an algebraic expression for the set of all points for which the sum of thedistances to A = (1, 0) and B = (−1, 0) is equal to 3. Answer: Square the equationq(x − 1)2+ y2+q(x + 1)2+ y2= 3, separate the remaining single square root on one sideand square again. Simplification gives 20x2+ 36y2= 45 which is equivalent tox2a2+y2b2= 1,where a, b can be computed as follows: because P = (a, 0) satisfies this equation, d(P, A) +d(P, B) = (a −1) + (a + 1) = 3 so that a = 3/2. Similarly, the point Q = (0, b) satisfying itgives d(Q, A) + d(P, B) = 2√b2+ 1 = 3 or b =√5/2.Here is a verification with the computer algebra system Mathematica. Writing L = d(P, A)and M = d(P, B) we simplify the equation L2+ M2= 32. The pa rt without square root is((L + M)2+ (L − M)2)/2 − 32. The remaining square root is ((L + M)2− (L − M)2)/2.Now square both and set them equal to see the equation 20x2+ 36y2= 45.L=Sqrt [ ( x−1)ˆ2+y ˆ 2 ] ; M=Sqrt [ ( x+1)ˆ2+y ˆ 2 ] ;Simplify [ ( ( ( L+M) ˆ 2 + ( L−M)ˆ2)/2 −3ˆ2)ˆ2 == ( ( ( L+M)ˆ2 −(L−M) ˆ 2 ) / 2 ) ˆ 2 ] The completion of the square of an equation x2+ bx + c = 0 is the idea to add(b/2)2− c on both sides to get (x + b/2 )2= (b/2)2− c. Solving for x gives thesolution x = −b/2 ±q(b/2)2− c.7 The equation 2x2−10x + 12 = 0 is equivalent to x2+ 5x = −6. Adding (5/2)2on both sidesgives (x + 5/2)2= 1/4 so that x = 2 o r x = 3.8 The equation x2+ …
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