Lesson 9: Volume and Cavalieri's Principle
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Quadratic Approximation at a Stationary Point Let F(X, Y) Be a Given
Multivariable Calculus Grinshpan Quadratic approximation at a stationary point Let f(x; y) be a given function and let (x0; y0) be a point in its domain. Under proper differentiability conditions one has f(x; y) = f(x0; y0) + fx(x0; y0)(x − x0) + fy(x0; y0)(y − y0) 1 2 1 2 + 2 fxx(x0; y0)(x − x0) + fxy(x0; y0)(x − x0)(y − y0) + 2 fyy(x0; y0)(y − y0) + higher−order terms: Let (x0; y0) be a stationary (critical) point of f: fx(x0; y0) = fy(x0; y0) = 0. Then 2 2 f(x; y) = f(x0; y0) + A(x − x0) + 2B(x − x0)(y − y0) + C(y − y0) + higher−order terms, 1 1 1 1 where A = 2 fxx(x0; y0);B = 2 fxy(x0; y0) = 2 fyx(x0; y0), and C = 2 fyy(x0; y0). Assume for simplicity that (x0; y0) = (0; 0) and f(0; 0) = 0. [ This can always be achieved by translation: f~(x; y) = f(x0 + x; y0 + y) − f(x0; y0). ] Then f(x; y) = Ax2 + 2Bxy + Cy2 + higher−order terms. Thus, provided A, B, C are not all zero, the graph of f near (0; 0) resembles the quadric surface z = Ax2 + 2Bxy + Cy2: Generically, this quadric surface is either an elliptic or a hyperbolic paraboloid. We distinguish three scenarios: * Elliptic paraboloid opening up, (0; 0) is a point of local minimum. * Elliptic paraboloid opening down, (0; 0) is a point of local maximum. * Hyperbolic paraboloid, (0; 0) is a saddle point. It should certainly be possible to tell which case we are dealing with by looking at the coefficients A, B, and C, and this is the idea behind the Second Partials Test. -
Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
Calculus & Analytic Geometry
TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Quadratic Equations in 3-D Match each function to its graph 1. 9x2 + 36y2 +4z2 = 36 2. 4x2 +9y2 4z2 =0 − 3. 36x2 +9y2 4z2 = 36 − 4. 4x2 9y2 4z2 = 36 − − 5. 9x2 +4y2 6z =0 − 6. 9x2 4y2 6z =0 − − 7. 4x2 + y2 +4z2 4y 4z +36=0 − − 8. 4x2 + y2 +4z2 4y 4z 36=0 − − − cone • ellipsoid • elliptic paraboloid • hyperbolic paraboloid • hyperboloid of one sheet • hyperboloid of two sheets 24 TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Parametric Equations (§10.1) and Vector Functions (§13.1) Definition. If x and y are given as continuous function x = f(t) y = g(t) over an interval of t-values, then the set of points (x, y)=(f(t),g(t)) defined by these equation is a parametric curve (sometimes called aplane curve). The equations are parametric equations for the curve. Often we think of parametric curves as describing the movement of a particle in a plane over time. Examples. x = 2cos t x = et 0 t π 1 t e y = 3sin t ≤ ≤ y = ln t ≤ ≤ Can we find parameterizations of known curves? the line segment circle x2 + y2 =1 from (1, 3) to (5, 1) Why restrict ourselves to only moving through planes? Why not space? And why not use our nifty vector notation? 25 Definition. If x, y, and z are given as continuous functions x = f(t) y = g(t) z = h(t) over an interval of t-values, then the set of points (x,y,z)= (f(t),g(t), h(t)) defined by these equation is a parametric curve (sometimes called a space curve). -
Chapter 3 Quadratic Curves, Quadric Surfaces
Chapter 3 Quadratic curves, quadric surfaces In this chapter we begin our study of curved surfaces. We focus on the quadric surfaces. To do this, we also need to look at quadratic curves, such as ellipses. We discuss: Equations and parametric descriptions of the plane quadratic curves: circles, ellipses, ² hyperbolas and parabolas. Equations and parametric descriptions of quadric surfaces, the 2{dimensional ana- ² logues of quadratic curves. We also discuss aspects of matrices, since they are relevant for our discussion. 3.1 Plane quadratic curves 3.1.1 From linear to quadratic equations Lines in the plane R2 are represented by linear equations and linear parametric descriptions. Degree 2 equations also correspond to curves you undoubtedly have come across before: circles, ellipses, hyperbolas and parabolas. This section is devoted to these curves. They will reoccur when we consider quadric surfaces, a class of fascinating shapes, since the intersection of a quadric surface with a plane consists of a quadratic curve. Lines di®er from quadratic curves in various respects, one of which is that all lines look the same (only their position in the plane may di®er), but that quadratic curves may truely di®er in shape. 3.1.2 The general equation of a quadratic curve The general equation of a line in R2 is ax + by = c. When we also allow terms of degree 2 in the variables x and y, i.e., x2, xy and y2, we obtain quadratic equations like x2 + y2 = 1, a circle. ² x2 + 2x + y = 3, a parabola; probably you recognize it as such if it is rewritten in the ² form y = 3 x2 2x or y = (x + 1)2 + 4. -
Math 16C: Multivariate Calculus
MATH 16C: MULTIVARIATE CALCULUS Jes´us De Loera, UC Davis April 12, 2010 Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS 7.1: The 3-dimensional coordinate system Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS 2 2 2 Thus, d = (x2 x1) + (y2 y1) + (z2 z1) . EXAMPLE: − − − Find the distancep between ( 3, 2, 5) and (4, 1, 2). − − d = (( 3) 4)2 + (2 ( 1))2 + (5 2)2 = √49+9+9= √67. q − − − − − Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS Mid-point between two points What is the mid-point between two points (x1, y1, z1) and (x2, y2, z2)? Let (xm, ym, zm) be the mid-point. Then xm must be half-way x1+x1 bewteen x1 and x2, so xm = 2 . y1+y2 Also, ym must be half-way between y1 and y2, so ym = 2 Similarly, zm must be half-way between z1 and z2, so z1+z2 zm = 2 . Thus, x1 + x1 y1 + y2 z1 + z2 (xm, ym, zm)= , , . 2 2 2 E.g. What is the midpoint bewteen ( 3, 2, 5) and (4, 1, 2)? − ( 3) + 4 2 + 1 5 + 2 1 3 1 (xm, ym, zm)= − , , = , , . 2 2 2 2 2 2 Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS Question: If we fix a point (a, b, c), what is the set of all points at a distance r from (a, b, c) called? Answer: A SPHERE centered at (a, b, c). (x a)2 + (y b)2 + (z c)2 = r (x a)2+(y b)2+(z c)2 = r 2. -
Calculus III -- Fall 2008
Section 12.2: Quadric Surfaces Goals : 1. To recognize and write equations of quadric surfaces 2. To graph quadric surfaces by hand Definitions: 1. A quadric surface is the three-dimensional graph of an equation that can (through appropriate transformations, if necessary), be written in either of the following forms: Ax 2 + By 2 + Cz 2 + J = 0 or Ax 2 + By 2 + Iz = 0 . 2. The intersection of a surface with a plane is called a trace of the surface in the plane. Notes : 1. There are 6 kinds of quadric surfaces. Scroll down to get an idea of what they look like. Keep in mind that each graph shown illustrates just one of many possible orientations of the surface. 2. The traces of quadric surfaces are conic sections (i.e. a parabola, ellipse, or hyperbola). 3. The key to graphing quadric surfaces is making use of traces in planes parallel to the xy , xz , and yz planes. 4. The following pages are from the lecture notes of Professor Eitan Angel, University of Colorado. Keep scrolling down (or press the Page Down key) to advance the slide show. Calculus III { Fall 2008 Lecture{QuadricSurfaces Eitan Angel University of Colorado Monday, September 8, 2008 E. Angel (CU) Calculus III 8 Sep 1 / 11 Now we will discuss second-degree equations (called quadric surfaces). These are the three dimensional analogues of conic sections. To sketch the graph of a quadric surface (or any surface), it is useful to determine curves of intersection of the surface with planes parallel to the coordinate planes. -
Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives
Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives By Andrew Grossfield, Ph. D., P. E. Abstract Many engineering students encounter and algebraically manipulate partial derivatives in their fluids, thermodynamics or electromagnetic wave theory courses. However it is possible that unless these students were properly introduced to these symbols, they may lack the insight that could be obtained from a geometric or visual approach to the equations that contain these symbols. We accept the approach that just as the direction of a curve at a point in two-dimensional space is described by the slope of the straight line tangent to the curve at that point, the orientation of a surface at a point in 3-dimensional space is determined by the orientation of the plane tangent to the surface at that point. A straight line has only one direction described by the same slope everywhere along its length; a tilted plane has the same orientation everywhere but has many slopes at each point. In fact the slope in almost every direction leading away from a point is different. How do we conceive of these differing slopes and how can they be evaluated? We are led to conclude that the derivative of a multivariable function is the gradient vector, and that it is wrong and misleading to define the gradient in terms of some kind of limiting process. This approach circumvents the need for all the unnecessary delta-epsilon arguments about obvious results, while providing visual insight into properties of multi-variable derivatives and differentials. This paper provides the visual connection displaying the remarkably simple and beautiful relationships between the gradient, the directional derivatives and the partial derivatives. -
Multivariable Calculus Math 21A
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. -