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:

For the scalar function Show 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 x=0, y=1?

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

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

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.

Calculate the vorticity for the flow field 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