Surface Integrals.

Slides:

Advertisements

Similar presentations

Section 18.2 Computing Line Integrals over Parameterized Curves

Advertisements

Surface Area and Surface Integrals

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.

Semester 1 Review Unit vector =. A line that contains the point P(x 0,y 0,z 0 ) and direction vector : parametric symmetric.

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

Parametric Surfaces.

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

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

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

Assigned work: pg. 449 #1abc, 2abc, 5, 7-9,12,15 Write out what “you think” the vector, parametric a symmetric equations of a line in R 3 would be.

MA Day 7 – January 15, 2013 Review: Dot and Cross Products Section 9.5 – Lines and Planes.

Surface Integral.

Center of Mass and Moments of Inertia

Vector Analysis Copyright © Cengage Learning. All rights reserved.

Vector Analysis 15 Copyright © Cengage Learning. All rights reserved.

Chapter 17 Section 17.8 Cylindrical Coordinates. x y z r The point of using different coordinate systems is to make the equations of certain curves much.

Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to the plane. It can be found by the Cross product of any.

1.3 Lines and Planes. To determine a line L, we need a point P(x 1,y 1,z 1 ) on L and a direction vector for the line L. The parametric equations of a.

Vector-Valued Functions. Step by Step: Finding Domain of a Vector-Valued Function 1.Find the domain of each component function 2.The domain of the vector-valued.

Chapter 15 Vector Analysis. Copyright © Houghton Mifflin Company. All rights reserved.15-2 Definition of Vector Field.

SECTION 13.7 SURFACE INTEGRALS. P2P213.7 SURFACE INTEGRALS  The relationship between surface integrals and surface area is much the same as the relationship.

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.

Equations of Lines and Planes

Sec 15.6 Directional Derivatives and the Gradient Vector

MA Day 61 – April 12, 2013 Pages : Tangent planes to parametric surfaces – an example Section 12.6: Surface area of parametric surfaces.

Double Integrals over Rectangles

Chapter 14 Multiple Integration. Copyright © Houghton Mifflin Company. All rights reserved.14-2 Figure 14.1.

Chapter 16 – Vector Calculus

Centers of Mass Review & Integration by Parts. Center of Mass: 2-Dimensional Case The System’s Center of Mass is defined to be:

9.13 Surface Integrals Arclength:.

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.

Section 17.7 Surface Integrals. Suppose f is a function of three variables whose domain includes the surface S. We divide S into patches S ij with area.

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.

Applying Multiple Integrals Thm. Let ρ(x,y) be the density of a lamina corresponding to a plane region R. The mass of the lamina is.

Copyright © Cengage Learning. All rights reserved.

CSE 681 Brief Review: Vectors. CSE 681 Vectors Basics Normalizing a vector => unit vector Dot product Cross product Reflection vector Parametric form.

12.10 Vectors in Space. Vectors in three dimensions can be considered from both a geometric and an algebraic approach. Most of the characteristics are.

CSE 681 Brief Review: Vectors. CSE 681 Vectors Direction in space Normalizing a vector => unit vector Dot product Cross product Parametric form of a line.

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.

The Divergence Theorem

Electric Flux Density, Gauss’s Law, and Divergence

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 =

Lesson 12 – 10 Vectors in Space

LESSON 90 – DISTANCES IN 3 SPACE

VECTORS APPLICATIONS NHAA/IMK/UNIMAP.

Intersection between - Lines, - Planes and - a plane & a Line

Lines and Planes in Space

Curl and Divergence.

Triple Integrals.

Copyright © Cengage Learning. All rights reserved.

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 =

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. {image}

Which of the equations below is an equation of a cone?

Radiative Field (Hubeny & Mihalas Chapter 3)

Introduction to Parametric Equations and Vectors

Find a vector equation for the line through the points {image} and {image} {image}

Find a vector equation for the line through the points {image} and {image} {image}

Some Theorems Thm. Divergence Theorem

17 VECTOR CALCULUS.

Copyright © Cengage Learning. All rights reserved.

Which of the equations below is an equation of a cone?

Copyright © Cengage Learning. All rights reserved.

Lines and Planes Ch13.5.

Find the directional derivative of the function at the given point in the direction of the vector v. {image}

Evaluate the line integral. {image}

17.5 Surface Integrals of Vector Fields

Evaluate the line integral. {image}

Presentation transcript:

Surface Integrals

Evaluating a Surface Integral Let S be a surface with equation z = g(x, y) and let R be its projection unto the xy-plane. If g, gx, and gy are continuous on R and f is continuous on S, then the surface integral of f over S is:

1. Evaluate the surface integral (Similar to p.1122 #1-4)

2. Evaluate the surface integral (Similar to p.1122 #1-4)

3. Evaluate the surface integral (Similar to p.1122 #5-6)

4. Find the mass of the surface lamina S of density ρ (Similar to p

Surface Integrals (parametric form)

5. Evaluate the integral (Similar to p.1122 #13-16)

6. Evaluate the integral (Similar to p.1122 #17-22)

Evaluating a Flux Integral Let S be an oriented surface given by z = g(x, y) and let R be its projection onto the xy-plane

7. Find the flux of F through S where N is the upward unit normal vector to S (Similar to p.1122 #23-28)

Summary: Surface Integrals [z = g(x,y)]

Summary: Surface Integrals [parametric form]