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 =

Slides:

Advertisements

Similar presentations

Surface Area and Surface Integrals

Advertisements

Tangent Planes and Linear Approximations

Teorema Stokes Pertemuan

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.

Theorems on divergent sequences. Theorem 1 If the sequence is increasing and not bounded from above then it diverges to +∞. Illustration =

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.

Chapter 15 – Multiple Integrals 15.7 Triple Integrals 1 Objectives:  Understand how to calculate triple integrals  Understand and apply the use of triple.

Chapter 12-Multiple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

Surface Integral.

Double Integrals over General Regions. Double Integrals over General Regions Type I Double integrals over general regions are evaluated as iterated integrals.

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

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

Section 16.3 Triple Integrals. A continuous function of 3 variable can be integrated over a solid region, W, in 3-space just as a function of two variables.

Vector Analysis Copyright © Cengage Learning. All rights reserved.

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

Double Integrals over Rectangles

Multiple Integration Copyright © Cengage Learning. All rights reserved.

By: Mohsin Tahir (GL) Waqas Akram Rao Arslan Ali Asghar Numan-ul-haq

Section 16.3 Triple Integrals. A continuous function of 3 variables can be integrated over a solid region, W, in 3-space just as a function of two variables.

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.

CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.1VECTOR FIELDS 14.2LINE INTEGRALS.

Triple Integrals in Spherical and Cylindrical In rectangular coordinates: dV = dzdydx In cylindrical coordinates: dV = r dzdrdθ In spherical coordinates:

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.

Graphing in 3-D Graphing in 3-D means that we need 3 coordinates to define a point (x,y,z) These are the coordinate planes, and they divide space into.

The Divergence Theorem

Lecture 19 Flux in Cartesian Coordinates.

Electric Flux Density, Gauss’s Law, and Divergence

Integration in Vector Fields

Find the area of the part of the plane 20x + 5y + z = 15 that lies in the first octant. Select the correct answer. The choices are rounded to the nearest.

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.

Triple Integrals.

Copyright © Cengage Learning. All rights reserved.

Use cylindrical coordinates to evaluate {image} where E is the solid that lies between the cylinders {image} and {image} above the xy-plane and below the.

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

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 =

Math 200 Week 7 - Friday Double Integrals.

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

rectangular coordinate system spherical coordinate system

Chapter 3 1. Line Integral Volume Integral Surface Integral

Evaluate the following integral: {image}

Triple Integrals Ex. Evaluate where.

Surface Integrals.

Some Theorems Thm. Divergence Theorem

Evaluate the iterated integral. {image} Select the correct answer

Chapter 15 Multiple Integrals

Christopher Crawford PHY

Find the volume of the solid obtained by rotating about the x-axis the region under the curve {image} from x = 2 to x = 3. Select the correct answer. {image}

Copyright © Cengage Learning. All rights reserved.

Find the area of the part of the plane 8x + 2y + z = 7 that lies inside the cylinder {image}

17 VECTOR CALCULUS.

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

13 Functions of Several Variables

Use Stokes' Theorem to evaluate {image} {image} S is a part of the hemisphere {image} that lies inside the cylinder {image} oriented in the direction of.

Calculate the iterated integral. {image}

Use Green's Theorem to evaluate the double integral

Copyright © Cengage Learning. All rights reserved.

Evaluate the line integral. {image}

Functions of Several Variables

Section Volumes by Slicing

DEPARTMENT OF PHYSICS GOVT.PG COLLEGE RAJOURI

15.7 Triple Integrals.

Evaluate the line integral. {image}

Presentation transcript:

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 = 3, z = 0, z = 5. -360 40 600 1,080 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Use the Divergence Theorem to calculate the surface integral {image} {image} S is the surface of the unit ball {image} {image} 1. 2. 3. 4. 5. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Use the Divergence Theorem to calculate the surface integral {image} that is, calculate the flux of F across S. {image} S is the sphere {image} {image} 1. 2. 3. 4. 5. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Use the Divergence Theorem to calculate the surface integral {image} that is, calculate the flux of F across S. {image} S is the surface of the solid that lies above the xy-plane and below the surface {image} 25.43 26.85 23.49 18.63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Let {image} Find the flux of F across the part of the paraboloid {image} that lies above the plane z = 1 and is oriented upward. {image} 1. 2. 3. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50