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 – Partial Derivative Definition PowerPoint presentation | free to download - id: 125f2c-Nzk3M

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Partial Derivative - Definition

- For a multi-dimensional scalar function, f, the

partial derivative with respect to a given

dimension at a specific point is defined as

follows - Backward six notation derivative notation

indicates that we are varying in the direction

indicated in the denominator while holding all

other variables constant

Higher order partial derivatives

- Partial derivatives can be applied multiple times

on a scalar function or vector. - The following are the four possibilities for the

second order partial derivatives of a function - Are and equal?

Mixed Derivative Theorem

- If a function f(x,y) is continuous and smooth to

second order, then the order of operation of the

partial derivatives does not matter and

Exercise

- For the function
- Show

Del Operator

- The del operator is a linear independent

combination of spatial partial derivatives. - In rectangular coordinates, it is expressed as
- (Notice the vector arrow in the second equality

is implied.) - Important!! The del operator always operates in

on a scalar or vector function to the right of it

within the term. The del operator is not

commutative like normal multiplication. Do NOT

apply the del operator to objects to its left.

Gradient Operator

- Application of the del operator to a scalar

function, f(x,y,z) is the same as taking the

gradient of a scalar function. The gradient is

defined - in rectangular coordinates as
- Notice that the result is a linear combination of

components and basis vectors and therefore the

gradient of a scalar function is a vector. - Since the result above is a vector, it obeys all

the rules from chapter 2 - ---We will only take gradients of scalar

functions in this course. It is possible to take

gradients of vectors but you obtain a 9-element

matrix called a Dyadic product

Gradient Operator

- In index notation, we establish the following

relationships for the del operator - The equality then takes the

following form in index notation

Exercise

- For the scalar function
- Show

Exercise

- Given a velocity field
- and the gradient of a scalar function
- expand the following expression

Gradient magnitude

- As we pointed out, since is a

vector, it has an associated magnitude and

direction. - To find the gradient magnitude use the definition

from chapter 2

Gradient direction

- Determining the direction of is a bit

more difficult. - Although we can use the definition from Chapter 2
- A geometric interpretation is more appropriate.
- Consider the differential of f (The differential

means a small change in the value of f ) - If we define a vector line element as
- The above differential can also be expressed as

Gradient direction

- Recall the geometric definition of the dot

product from chapter 2 - Where q is the coplanar angle between the vector

and - The above expression indicates that df is maximum

when is parallel to the gradient (when q is

0) - Therefore the above expression also shows that f

increases most rapidly when is in the

direction of or that is in the

direction that causes the biggest change in f. - The direction of is called the

ascendant of f

Advection

- We can now measure the change a scalar quantity

along any direction. - For example, we can find the change in the

arbitrary scalar f(x,y,z) in a general unit

direction by taking the dot

product of with - The time dependence in the above expression comes

about due to considering a parameterized set of

curves

Advection

- Take the derivative of f with respect to t and

use the chain rule. We can thus see how we obtain

- The expression on the right gives the variation

of f in the direction of - (Notice that if is parallel to , the

expression on right is the gradient of f

reiterating the last proof regarding the

direction of the gradient. )

Advection

- In meteorology and oceanography we are often

interested in the rate of change of a physical

quantity along the direction of the flow field, - For example, the rate of change of f along the

flow field is - The term on the right is related to a physical

quantity called advection and is one of two

contributions to the total or material

derivative. We will learn more about advection

and the material derivate in the next chapter.

Advection

- Notice that if
- We observe that the function, f, is constant

along the direction of the flow field. This

comes up often in oceanography and meteorology. - For example, for pressure field, p, what does it

mean if ?

Exercise

- For the scalar function
- Find the magnitude and direction of the vector
- What do you expect equals at x0,

y1?

Divergence

- There are two common ways to apply a del operator

on a vector the divergence and the curl. The

divergence operation results in a scalar quantity

while the curl results in a vector quantity. - For a vector field
- The Divergence on is defined as
- In index notation the divergence takes the form

Divergence physical interpretation

- From a physical standpoint, the divergence is a

measure of the addition or removal of a vector

quantity. A system with positive divergence is

called a source. A system with negative

divergence is called a sink. A system with no

divergence, is called solenoidal or

divergenceless

Exercise

- For the flow field
- Calculate at
- x0, y1/2
- X1/2, y0

Curl

- For a vector field
- The curl on is defined as

Curl Physical interpretation

- Physically the curl is a measure of the

rotational properties of a vector about a point.

- For fluid field ,the curl is measure of

the rotation of a fluid parcel about its center

of mass and is called the vorticity denoted by

the vector omega . - If the fluid vorticity is zero
- it is considered irrotational.

Curl

- In meteorology and oceanography, one is often

interested in the vertical vorticity component - This vertical vorticity component is a measure of

the horizontal shear of the medium.

Exercise

- Calculate the vorticity for the flow field

Laplacian of a scalar

- For certain velocity fields (irrotational), it is

possible to relate the velocity field vector to a

scalar quantity called the velocity potential - If we wish to examine the divergence of this

unique velocity field, we obtain a second order

partial differential operator on f called the

Laplacian of f - In index notation, the Laplacian takes the form

Laplacian of a scalar

- As a general operator, the Laplacian is defined

in rectangular coordinates as - The Laplacian can be applied to either a scalar

or vector - - If applied to a scalar the results is a scalar
- - If applied to a vector, the result is a vector

Laplacian of a scalar

- In 1-D calculus we found the max and min of a

function, f(x), by finding at which points - We could then resolve if the point was a max or

min by whether the second derivative was less

than or greater than 0 respectively. - Similarly, the Laplacian of a scalar allows us to

determine wether the local extrema of a

multivariable function is a 1)maximum - - 2) Minimum -
- 3) Saddle Point -

Exercise

- For the scalar function
- Locate the extrema.
- Determine if your point(s) is/are a maximum or

minimum

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 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Partial Derivative Definition" 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