View by Category

The presentation will start after a short

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

Loading...

PPT – Taylors series PowerPoint presentation | free to download - id: 37f0c-MmRhY

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Taylors series

- Before going into Gauss Theorem and Mass

conservation it will be useful to review Taylor

series. The concept of expanding a function about

a point will be used many times throughout this

course and in SO 414. - If a function converges about the point xo , we

can represent the function as a power series - Note The term means terms of

4th order and higher.

Taylors series

- The coefficients are

evaluated by examining the derivatives of f(x) at

xxo - where

Taylors series

- Substitution of the preceding coefficient values

into the power series expansion yields the Taylor

series of the function - Notice there are no formal restrictions on the

selection of the point xo besides the requirement

that the function must converge. - There are some practical limitations however such

as choosing xo near the center of the domain, so

that we dont have to consider an overwhelming

number of terms in the Taylor expansion.

Taylors series

- To ensure the series converges in a limited

number of terms, we will restrict the domain such

that - Then we can substitute in

the Taylor series where - The Taylor series is then expressed as
- The advantage of this form is we know the series

converges and we only need to consider a couple

terms

Taylors series

- Since it is safe to assume accuracy with only a

couple terms we will approximate the Taylor

Series as

Taylors series

- In rectangular coordinates, the multivariable

Taylor expansion is fairly straightforward. By

considering variations in only one variable at a

time, we are essentially looking at three

different 1-D cases of the previous equation. - If we consider variation in all three direction,

we obtain - With and

Gauss Theorem

- Now we utilize the Taylor series to derive a

theorem that will be of great value to us. - Gauss Theorem
- Given a fixed volume V which is bounded by the

area A and unit vector normal to the

surface element dA - For a uniformly continuous vector field,
- Gauss theorem states

Gauss Theorem - proof

- Consider an infinitesimal cube with sides dx,dy

and dz as shown where the center of the cube is

at position xo, yo and zo.

Gauss Theorem - proof

- Now examine the flow out of each face of the cube.

Gauss Theorem - proof

- Now use Taylor series expansions on all the

terms. - For example, the first two terms have Taylor

series expansions - Adding these two terms together we obtain

Gauss Theorem - proof

- Performing the same analysis on the y and z cube

faces we obtain - and

Gauss Theorem - proof

- Combining all of the above expressions we obtain

the result - Since the above result was performed on an

infinitesimal cube, we can sum up (or Riemann

integrate) all the cubes that make up an

arbitrary shape. - In doing such a summation we observe that the

vector contributions of adjacent faces are equal

and opposite to each other and cancel each other

out. Such cancellations will take place

throughout the entire interior so that we are

left only with the contributions through the

surface of our arbitrary shape.

Gauss Theorem - proof

- The result after taking the summation under the

limit of an infinite number of cubes whose volume

approaches zero is Gauss Theorem - A more common notation convention is to imply the

triple and double integrals over the volume and

surface area respectively

Exercise Archimedes Principle

- The net force due to hydrostatic pressure on a

submerged object is mathematically described as - Where is the outward normal to the

submerged object. - You may assume the medium is incompressible so
- rwater constant
- Use Gauss theorem to show that the above net

buoyant force of the surface of the object is

also equal to the weight of the displaced water

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2015 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2015 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Taylors series" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Usna University and other schools with their online training by sharing educational presentations for free