VC.10 Surface Area Calculations and Surface Integrals (Day 2)

Slides:



Advertisements
Similar presentations
Differential Calculus (revisited):
Advertisements

Section 18.2 Computing Line Integrals over Parameterized Curves
Surface Area and Surface Integrals
Area between curves. How would you determine the area between two graphs?
Chapter 9: Vector Differential Calculus Vector Functions of One Variable -- a vector, each component of which is a function of the same variable.
Teorema Stokes Pertemuan
EE3321 ELECTROMAGENTIC FIELD THEORY
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 18 Section 18.5 Green’s Theorem. Closed Curves and Simple Closed Curves Closed curves are figures that can be drawn so that you begin and end.
Double Integrals Area/Surface Area Triple Integrals.
VECTOR CALCULUS VECTOR CALCULUS The main results of this chapter are all higher-dimensional versions of the Fundamental Theorem of Calculus (FTC).
Section 16.7 Surface Integrals. Surface Integrals We now consider integrating functions over a surface S that lies above some plane region D.
VECTOR CALCULUS The Divergence Theorem In this section, we will learn about: The Divergence Theorem for simple solid regions, and its applications.
Stokes’ Theorem Divergence Theorem
Chapter 16 – Vector Calculus 16.9 The Divergence Theorem 1 Objectives:  Understand The Divergence Theorem for simple solid regions.  Use Stokes’ Theorem.
PHYSICS-II (PHY C132) ELECTRICITY & MAGNETISM
1 April 14 Triple product 6.3 Triple products Triple scalar product: Chapter 6 Vector Analysis A B C + _.
Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.
Vector Analysis Copyright © Cengage Learning. All rights reserved.
MA Day 55 – April 4, 2013 Section 13.3: The fundamental theorem for line integrals – Review theorems – Finding Potential functions – The Law of.
Copyright © Cengage Learning. All rights reserved.
Notes 13 ECE 2317 Applied Electricity and Magnetism Prof. D. Wilton
Chapter 15 Vector Analysis. Copyright © Houghton Mifflin Company. All rights reserved.15-2 Definition of Vector Field.
§1.2 Differential Calculus
Copyright © Cengage Learning. All rights reserved. Vector Analysis.
§1.2 Differential Calculus Christopher Crawford PHY 416G
Chapter 16 – Vector Calculus 16.1 Vector Fields 1 Objectives:  Understand the different types of vector fields Dr. Erickson.
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 14 Vector Calculus.
Parametric Surfaces We can use parametric equations to describe a curve. Because a curve is one dimensional, we only need one parameter. If we want to.
Also known as Gauss’ Theorem
CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.1VECTOR FIELDS 14.2LINE INTEGRALS.
CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.1VECTOR FIELDS 14.2LINE INTEGRALS.
§1.3 Integrals Flux, Flow, Subst Christopher Crawford PHY
CHAPTER 9.10~9.17 Vector Calculus.
Vector Analysis 15 Copyright © Cengage Learning. All rights reserved.
1 Line Integrals In this section we are now going to introduce a new kind of integral. However, before we do that it is important to note that you will.
Vector integration Linear integrals Vector area and surface integrals
Chapter 6 Vector Analysis
The Divergence Theorem
Integration in Vector Fields
Use the Divergence Theorem to calculate the surface integral {image} {image} S is the surface of the box bounded by the planes x = 0, x = 4, y = 0, y =
Chapter 9 Vector Calculus.
1 Divergence Theorem. 2 Understand and use the Divergence Theorem. Use the Divergence Theorem to calculate flux. Objectives Total flux change = (field.
Curl and Divergence.
Use Green's Theorem to evaluate the double integral
Use the Divergence Theorem to calculate the surface integral {image} {image} S is the surface of the box bounded by the planes x = 0, x = 2, y = 0, y =
Chapter 3 1. Line Integral Volume Integral Surface Integral
Some Theorems Thm. Divergence Theorem
Math 265 Created by Educational Technology Network
Chapter 6 Vector Analysis
§1.3 Integrals Flux, Flow, Subst
Copyright © Cengage Learning. All rights reserved.
17 VECTOR CALCULUS.
Copyright © Cengage Learning. All rights reserved.
Use Green's Theorem to evaluate the double integral
Copyright © Cengage Learning. All rights reserved.
Applications of Integration
Evaluate the line integral. {image}
Copyright © Cengage Learning. All rights reserved.
DEPARTMENT OF PHYSICS GOVT.PG COLLEGE RAJOURI
Applications of Integration
Evaluate the line integral. {image}
Presentation transcript:

VC.10 Surface Area Calculations and Surface Integrals (Day 2)

Example 1: Surface Area

The Surface Area Conversion Factor

A longer normal vector means that the uv-rectangle has much more surface area in xyz-space (more curvature, really)

The Surface Area Conversion Factor

Example 1 Revisited: Surface Area

Example 2: Surface Integrals

Summary: Surface Integrals:

Example 3: Surface Area Enclosed in a Curve Plotted on a Surface (Bonus Material!)

Today’s Material Did Not Reference Vector Fields Acting on a Surface! Let R be a solid in three dimensions with boundary surface (skin) C with no singularities on the interior region R of C. Then the net flow of the vector field Field(x,y,z) ACROSS the closed surface is measured by:

Today’s Material Did Not Reference Vector Fields Acting on a Surface! The Divergence Theorem is great for a closed surface, but it is not useful at all when your surface does not fully enclose a solid region. In this situation, we will need to compute a surface integral. For a parameterized surface, this is pretty straightforward:

Feedback: Privacy Policy Feedback
Do Not Sell My Personal Information
About project: SlidePlayer Terms of Service

© 2024 SlidePlayer.com Inc. All rights reserved.