METEO 300
Fundamentals of Atmospheric Science

8.1 This is why partial derivatives are so easy...

Extra Credit Reminder!

Here is another chance to earn one point of extra credit: Picture of the Week!

  1. You take a picture of some atmospheric phenomenon - a cloud, wind-blown dust, precipitation, haze, winds blowing different directions - anything that strikes you as interesting.
  2. Add a short description of the processes that you think are causing your observation. A Word file is a good format for submission.
  3. Use your name as the name of the file. Upload it to the Picture of the Week Dropbox in this week's lesson module. To be eligible for the week, your picture must be submitted by 23:59 UT on Sunday of each week.
  4. I will be the sole judge of the weekly winners. A student can win up to three times.
  5. There will be a Picture of the Week dropbox each week through Lesson 11. Keep submitting entries!

In your first calculus class you learned about derivatives. Suppose we have a function f that is a function of x, which we can write as f(x). What is the derivative of f with respect to x?

df(x) dx This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.1]

What about a new function that depends on two variables, h(x,y)? This function could, for example, give the height h of mountainous terrain for each horizontal point (x,y). So what is the derivative of h with respect to x? One way we determine this derivative is to fix the value of y = y1, which is the same as assuming that y is a constant, and then take the ordinary derivative of h with respect to x. In a sense, we are taking a slice through the mountain in the x-direction at a fixed value of y=y1. Thus,

( dh dx ) y=constant h x   This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.2]

This is called the partial derivative of h with respect to x. It’s pretty easy to determine because we do not need to worry about how y might depend on x.

Check Your Understanding

Let h = (x-3)2cos(y). What is the partial derivative of h with respect to x?

Click for answer.

h x = ( ( x3 ) 2 cos(y) ) x =2( x3 )cos(y) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

We can also find the partial derivative of h with respect to y. Can you do this?

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h y = ( ( x3 ) 2 cos(y) ) y = ( x3 ) 2 sin(y) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

So you can see that the h / x This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. may be different for each value of y and h / y This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. may be different for each value of x. Thus, even if you are not entirely familiar with partial derivatives and their notation, you can see that they are no different from ordinary derivatives but you take the derivative for just of one variable at a time.