2006 Fall MATH 100 Lecture 221 MATH 100 Lecture 22 Introduction to surface integrals.

Slides:

Advertisements

Similar presentations

Surface Area and Surface Integrals

Advertisements

The gradient as a normal vector. Consider z=f(x,y) and let F(x,y,z) = f(x,y)-z Let P=(x 0,y 0,z 0 ) be a point on the surface of F(x,y,z) Let C be any.

Tangent Vectors and Normal Vectors. Definitions of Unit Tangent Vector.

Stokes Theorem. Recall Green’s Theorem for calculating line integrals Suppose C is a piecewise smooth closed curve that is the boundary of an open region.

Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives:  Understand integration of different types of surfaces Dr. Erickson.

Honors Appendix Advanced Integrated Math I. Warm-Up: April 20, 2015.

Rates of Change and Tangent Lines Section 2.4. Average Rates of Change The average rate of change of a quantity over a period of time is the amount of.

The relation between the induced charge density σ p and the polarisation, P is Where is a unit vector along the outward normal to the surface. P is along.

Are you attending lecture today? 1.Yes 2.No

EEE 340Lecture Spherical Coordinates. EEE 340Lecture 042 A vector in spherical coordinates The local base vectors from a right –handed system.

2-7 Divergence of a Vector Field

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

Section 16.7 Surface Integrals. Surface Integrals We now consider integrating functions over a surface S that lies above some plane region D.

02-1 Physics I Class 02 One-Dimensional Motion Definitions.

Lecture#6: segmentation Anat Levin Introduction to Computer Vision Class Fall 2009 Department of Computer Science and App math, Weizmann Institute of Science.

2006 Fall MATH 100 Lecture 141 MATH 100Class 21 Line Integral independent of path.

Stokes’ Theorem Divergence Theorem

2006 Fall MATH 100 Lecture 141 MATH 100 Class 20 Line Integral.

2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 25 Final review Class 25 Final review 1.Function of two or more variables.

X y z Point can be viewed as intersection of surfaces called coordinate surfaces. Coordinate surfaces are selected from 3 different sets.

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

Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.

Optics and Light Lenses form images by refracting light.

The integral of a vector field a(x) on a sphere Letbe a vector field. Here are coordinate basis vectors, whilstare local orthonormal basis vectors. Let.

Lecture 9 Dustin Lueker.  Can not list all possible values with probabilities ◦ Probabilities are assigned to intervals of numbers  Probability of an.

8.8 Improper Integrals Math 6B Calculus II. Type 1: Infinite Integrals  Definition of an Improper Integral of Type 1 provided this limit exists (as a.

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.

Vector Calculus CHAPTER 9.10~9.17. Ch9.10~9.17_2 Contents  9.10 Double Integrals 9.10 Double Integrals  9.11 Double Integrals in Polar Coordinates 9.11.

The Cut Cone. The given views show the Front Elevation and part Plan of a cut cone. Draw the following views :- Complete Plan End Elevation Development.

8.2 Area of a Surface of Revolution

Electric Field Define electric field, which is independent of the test charge, q, and depends only on position in space: dipole One is > 0, the other

Vector Valued Functions

2006: Assoc. Prof. R. J. Reeves Gravitation 2.1 P113 Gravitation: Lecture 2 Gravitation near the earth Principle of Equivalence Gravitational Potential.

Further Applications of Green’s Theorem Ex. 2 Ex. 2) Area of a plane region -for ellipse: -in polar coordinates: Ex. 4 Ex. 4) Double integral of the Laplacian.

Lecture 4-1 At a point P on axis: At a point P off axis: At point P on axis of ring: ds.

Source Page US:official&tbm=isch&tbnid=Mli6kxZ3HfiCRM:&imgrefurl=

Section 9.2 Area of a Surface of Revolution. THE AREA OF A FRUSTUM The area of the frustum of a cone is given by.

Путешествуй со мной и узнаешь, где я сегодня побывал.

Lecture 3-1 Electric Field Define electric field, which is independent of the test charge, q, and depends only on position in space: dipole One is > 0,

(MTH 250) Lecture 19 Calculus. Previous Lecture’s Summary Definite integrals Fundamental theorem of calculus Mean value theorem for integrals Fundamental.

University of Utah Introduction to Electromagnetics Lecture 14: Vectors and Coordinate Systems Dr. Cynthia Furse University of Utah Department of Electrical.

MA Day 62 – April 15, 2013 Section 13.6: Surface integrals.

Vector-Valued Functions and Motion in Space

Integrals gradient.

Electricity and Magnetism

(MTH 250) Calculus Lecture 22.

Curl and Divergence.

Copyright © Cengage Learning. All rights reserved.

Page 1. Page 2 Page 3 Page 4 Page 5 Page 6 Page 7.

Gauss’s Law W4D2.

How to calculate a dot product

Electricity and Magnetism

Properties of Gradient Fields

Literacy in Maths Gradient Tangent Normal

Surface Integrals.

5.4 – Applying Definite Integration

12/1/2018 Normal Distributions

13.7 day 2 Tangent Planes and Normal Lines

Extract Object Boundaries in Noisy Images

Arc Length and Surface Area

Lecture 2 Section 1.3 Objectives: Continuous Distributions

Creating Meshes Through Parameterized Functions

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Lecture 17 Today Divergence of a vector filed

Creating Meshes Through Parameterized Functions

Test # 1 Functional Analysis

17.5 Surface Integrals of Vector Fields

STA 291 Spring 2008 Lecture 8 Dustin Lueker.

Find the following limit. {image}

Presentation transcript:

2006 Fall MATH 100 Lecture 221 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 222 Definition of density function:

2006 Fall MATH 100 Lecture 223 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 224 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 225 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 226 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 227 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 228 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 229 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2210 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2211 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2212 Surface integral of vector functions, we have studied line (curve) integral with orientation, now we go to surface with orientation. In general, a surface is given by G(x,y,z) = 0 The particular cones are MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2213 There are 2 unit normal vectors A surface has 2 orientation, corresponding to the 2 normal direction. The orientation should be chosen in the way that there is no sudden change in the normal direction when we transverse along the surface. MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2214 The 2 possible orientation: inward normal and outward normal MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2215 Surface integral

2006 Fall MATH 100 Lecture 2216 MATH 100 Lecture 22 Introduction to surface integrals (continuous next page)

2006 Fall MATH 100 Lecture 2217 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2218 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2219 MATH 100 Lecture 22 Introduction to surface integrals

2006 Fall MATH 100 Lecture 2220 MATH 100 Lecture 22 Introduction to surface integrals