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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

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