Multivariable CalculusOliver KnillMath 21a, Fall 2011These notes contain condensed ”two pages per lecture” notes with essential informationonly. Remaining space was filled with problems.Harvard Multivariable Calculus Math 21a, Fall 2011Math 21a: Multivariable calculus Oliver Knill, Fall 20111: Geometry and DistanceA 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 }.1 Describe the location of the points P = (1, 2, 3), Q = (0, 0, −5), R = (1, 2, −3) in words.Possible Answer: P = (1, 2, 3) is in the positive octant of space, where all coordinatesare positive. The point R = (0, 0, −5) is on the negative z axis. The point S = (1, 2, −3) isbelow the xy-plane. When projected onto the xy-plane it is in the first quadrant.2 Problem. Find the midpoint M of P = (1, 2, 5) and Q = (−3, 4, 7). Answer. The midpointis obtained by taking the average of each coordinat e M = (P + Q)/2 = (−1, 3, 6 ) .The Euclidean distance between two p oints P =(x, y, z) and Q = (a, b, c) in space is defined as d(P, Q) =q(x −a)2+ (y − b)2+ (z −c)2.This definition of Euclidean distance is motivated by the Pythagorean theorem.13 Find the distance d(P, Q) between the points P = (1, 2, 5) and Q = (−3, 4, 7) and verify t hatd(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).A circle of radius r centered at P = (a, b) is the collection of points in the planewhich 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=ρ2.4 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 smaller than 4the radius of the sphere. The point is inside.1In an appendix to ”Geometry” of his ”Discours de la m´ethode” which appeared in 1637, Ren´e Descartes(1596- 1650). More about Descartes in ”Descartes Secret Notebook” by Amir Aczel.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.25 Find the roots of the quadrat ic equation 2x2− 10x + 12 = 0. Answer. The equation isequivalent to x2+ 5x = −6. Adding (5/2)2on both sides gives (x + 5/2)2= 1/4 so thatx = 2 or x = 3.6 Find the center of the sphere x2+ 5x + y2− 2y + z2= −1. Answer: Complete the squareto get (x + 5/ 2)2−25/4 + (y −1)2−1 + z2= −1 or (x −5/2)2+ (y −1)2+ z2= (5/2)2. Wesee a sphere center (5/2, 1, 0) and radius 5/2.Al-KhwarizaiRene DescartesDistance between spheres7 Find the set of points P = (x, y, z) in space which satisfy x2+ y2= 9. Answer: This is acylinder of radius 3 around t he z-axes parallel to the y axis.8 a)Find the distances of P = (12, 5, 0) to each of the 3 coordinate axes. b) Find the distanceof P = (1 2, 5, 0) to the coordinate planes: Answer a): 12, 5, 13. Answer b): 12, 5, 0.9 Find the center and radius of the sphere x2+ 2x + y2− 16y + z2+ 10z + 54 = 0. Answer:Do a completion of square (x + 2)2+ ( y −8)2+ ( z + 5 )2= 36 is the equation of the sphere.10 Describe the set xy = 0. Answer: We either must have x = 0 which is the yz-plane, ory = 0 which is the xz-plane. The set is a union of two planes.11 Find an equation for the set of points which have the same distance t o (1, 1, 1) and (0 , 0, 0).Answer: ( x −1)2+ (y −1)2+ (z − 1)2= x2+ y2+ z2gives −2x + 1 − 2y + 1 − 2z + 1 = 0or 2x + 2y + 2z = 3. We will see that this is the equation of a plane.12 Find the distance between the spheres x2+(y −12)2+z2= 1 and (x−3)2+y2+(z −4)2= 9.Answer:The distance between the centers is√32+ 42+ 122= 13. The distance betweenthe spheres is 13 − 3 − 1 = 9.2Due to Al- Khwarizmi (780-850) in ”Compendium on Calculation by Completion and Reduction” The book”The mathematics of Egypt, Mesopotamia,China, India and Islam, a Source b ook, Ed Victor Katz, containstranslations of some of this work.Math 21a: Multivariable calculus Oliver Knill, Fall 20112: Vectors and Dot productTwo points P = (a, b, c) and Q = (x, y, z) in space define a vector ~v = hx − a, y −b − z − ci. It points from P to Q and we write also ~v =~P Q. The real numbersnumbers p, q, r in a vector ~v = hv1, v2, v3i are called the components of ~v.Similar definitions hold in two dimensions, where vectors have two components. Vectors can bedrawn everywhere in space but two vectors with the same components are considered equal.1The addition of two vectors is ~u + ~v = hu1, u2, u3i + hv1, v2, v3i =hu1+v1, u2+v2, u3+v3i. The scalar multiple λ~u = λhu1, u2, u3i = hλu1, λu2, λu3i.The difference ~u −~v can best be seen as the addition of ~u and (−1) ·~v.The addition and scalar multiplication of vectors satisfy the laws you know from arithmetic.commutativity ~u +~v = ~v + ~u, associativity ~u + (~v + ~w) = (~u +~v) + ~w and r ∗(s ∗~v) = (r ∗s) ∗~vas well as distributivity (r+s)~v = ~v(r+s) and r(~v+ ~w) = r~v+r ~w, where ∗ is scalar multiplication.The length |~v| of a vector ~v =~P Q is defined as the distance d(P, Q) from P to Q.A vector of length 1 is called a unit vector.1 |h3, 4i| = 5 and |h3, 4, 12i| = 13. Examples of unit vectors are |~i| = |~j| =~k| = 1 andh3/5, 4/5 i and h3/13, 4/13, 1 2/13i. The only vector of length 0 is the zero vector |~0| = 0.The dot product of two vectors ~v = ha, b, ci and ~w = hp, q, ri is defined as ~v · ~w =ap + bq + cr.The dot product determines distance and distance determines the dot product.Proof: Lets write v = ~v in this proof. Using the dot product one can express the …
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