Transformations.

Slides:

Advertisements

Similar presentations

Lecture 5: Jacobians In 1D problems we are used to a simple change of variables, e.g. from x to u 1D Jacobian maps strips of width dx to strips of width.

Advertisements

Multiple Integral. Double Integrals over Rectangles Remark ：

Double Integrals Area/Surface Area Triple Integrals.

2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

Lecture 15 Today Transformations between coordinate systems 1.Cartesian to cylindrical transformations 2.Cartesian to spherical transformations.

For each point (x,y,z) in R3, the cylindrical coordinates (r,,z) are defined by the polar coordinates r and  (for x and y) together with z. Example Find.

MULTIPLE INTEGRALS MULTIPLE INTEGRALS 16.9 Change of Variables in Multiple Integrals In this section, we will learn about: The change of variables.

Recall :The area of the parallelogram determined by two 3-vectors is the length of cross product of the two vectors. A parametrized surface defined by.

Vector Products (Cross Product). Torque F r T F r T F1F1 F2F2.

Chapter 15 – Multiple Integrals

Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.

Multiple Integrals 12. Surface Area Surface Area In this section we apply double integrals to the problem of computing the area of a surface.

Lecture 18: Triple Integrals, Cyclindrical Coordinates, and Spherical Coordinates.

MA Day 46 – March 19, 2013 Section 9.7: Spherical Coordinates Section 12.8: Triple Integrals in Spherical Coordinates.

Lecture 19: Triple Integrals with Cyclindrical Coordinates and Spherical Coordinates, Double Integrals for Surface Area, Vector Fields, and Line Integrals.

MA Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.

Polar Coordinates. Common Coordinate Systems There are two common coordinate systems: Cartesian Rectangular Coordinate SystemPolar Coordinate System.

Surface Integral.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Double Integrals a. (16.2.1),

CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

VC.07 Change of Variables to Compute Double Integrals.

VECTOR CALCULUS. Vector Multiplication b sin   A = a  b Area of the parallelogram formed by a and b.

Sec 16.7 Triple Integrals in Cylindrical Coordinates In the cylindrical coordinate system, a point P is represented by the ordered triple (r, θ, z), where.

11.7 Cylindrical and Spherical Coordinates. The Cylindrical Coordinate System In a cylindrical coordinate system, a point P in space is represented by.

Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 13 Multiple Integration.

MA Day 60 – April 11, MA The material we will cover before test #4 is:

7-4 Translations and Vectors. U SING P ROPERTIES OF T RANSLATIONS PP ' = QQ ' PP ' QQ ', or PP ' and QQ ' are collinear. P Q P 'P ' Q 'Q ' A translation.

CP3 Revision: Calculus Basic techniques of differentiation and integration are assumed Taylor/MacLaurin series (functions of 1 variable) x y Situation.

Parametric Surfaces and their Area Part II. Parametric Surfaces – Tangent Plane The line u = u 0 is mapped to the gridline C 2 =r(u 0,v) Consider the.

ESSENTIAL CALCULUS CH12 Multiple integrals. In this Chapter: 12.1 Double Integrals over Rectangles 12.2 Double Integrals over General Regions 12.3 Double.

Consider a transformation T(u,v) = (x(u,v), y(u,v)) from R 2 to R 2. Suppose T is a linear transformation T(u,v) = (au + bv, cu + dv). Then the derivative.

SECTION 12.8 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS.

1.8 Glide Reflections and Compositions Warm Up Determine the coordinates of the image of P(4, –7) under each transformation. 1. a translation 3 units left.

14.7 Day 2 Triple Integrals Using Spherical Coordinates and more applications of cylindrical coordinates This is a Klein bottle, It is a 4 dimensional.

Wave Equations: EM Waves. Electromagnetic waves for E field for B field.

Copyright © Cengage Learning. All rights reserved Change of Variables in Multiple Integrals.

CALCULUS III CHAPTER 5: Orthogonal curvilinear coordinates

MA Day 58 – April 9, MA The material we will cover before test #4 is:

TRIPPLE INTEGRAL For a bounded closed 3-D area V and a function continuous on V, we define the tripple integral of with respect to V, in symbols in much.

Section 3.4 Cross Product.

Copyright © Cengage Learning. All rights reserved.

Chapter 3 Overview.

Vector-Valued Functions and Motion in Space

© 2010 Pearson Education, Inc. All rights reserved

Hire Toyota Innova in Delhi for Outstation Tour

The Jacobian & Change of variables

Copyright © Cengage Learning. All rights reserved.

課程大綱 OUTLINE Double Integrals（二重積分） Triple Integrals（三重積分）

Vector Products (Cross Product)

Change of Variables in Multiple Integrals

7.4 Translations and vectors

Find the Jacobian of the transformation. {image}

Chapter 15 Multiple Integrals

Copyright © Cengage Learning. All rights reserved.

Find the Jacobian of the transformation

Transformations –Translation

15.4 Double Integrals in Polar Form

UNIT V JACOBIANS Dr. T. VENKATESAN Assistant Professor

Multiple Integrals.

Maps one figure onto another figure in a plane.

9.2 Reflections By Brit Caswell.

Chapter 16: Double and Triple Integrals

Evaluate the integral {image}

Presentation transcript:

Transformations

Left boundary of S maps to r(u0,v) Lower boundary of S maps to r(u,v0) r(u,v) = < x(u,v), x(u,v) > is the position vector of the image of the point (u,v) The image of S under T is not a parallelogram - but can be approximated by a parallelogram spanned by the vectors a and b

The image of S under T is not a parallelogram - but can be approximated by a parallelogram spanned by the vectors a and b

Area

Jacobian

Polar Coordinates

Polar Coordinates

Triple Integrals

Spherical

Spherical