1. 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

2. 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?

3. 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

4. Exercise
For the function
Show

5. 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.

6. 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

7. Gradient Operator
In index notation, we establish the following relationships for the del operator:
The equality then takes the following form in index notation:

8. For the scalar function Show
Exercise
For the scalar function
Show

9. Exercise Given a velocity field and the gradient of a scalar function
expand the following expression

10. 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

11. 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

12. 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

13. 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

14. 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. )

15. 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.

16. 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 ?

17. Exercise For the scalar function
Find the magnitude and direction of the vector
What do you expect equals at x=0, y=1?

18. 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:

19. 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

20. For the flow field Calculate at: x=0, y=1/2 X=1/2, y=0
Exercise
For the flow field
Calculate at:
x=0, y=1/2
X=1/2, y=0

21. For a vector field The curl on is defined as

22. 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.

23. 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.

24. Calculate the vorticity for the flow field
Exercise
Calculate the vorticity for the flow field

25. 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

26. 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

27. 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 -

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