Looked up the topic on Google and only found course studies,
not a real definition.
Explain them to me.
What are Partial diferentials good for? Is there some value to a half-assed job? Why not FULLY differentiate them?
(Can you tell I don’t know calculus?)
Partial differential equations involve partial derivatives, rather than ordinary differential equations which involve regular derivatives. The difference is that the function you are attempting to solve for depends on more than one variable. Partial differential equations come up everywhere in engineering as well as in quantum mechanics. Its almost impossible to understand differential equations without at least a little knowledge of calculus.
A partial derivative (or differential) is the derivative of a function of more than one variable with respect to one of the variables. If you have F(x,y) a function of x and y, you can take the partial derivative with respect to either x or y. For instance, d(F(x,y))/dx is the derivative with respect to x and d(F(x,y))/dy is the derivative with respect to y.
F(x,y) = 3x + 2y
d(F(x,y))/dx = 3
d(F(x,y))/dy = 2
Basicly, you are getting the rate of change of the variable that you are differentiating with respect to.
What are they good for? Well when you are working with single variable functions, nothing. If, however, you are working with multi-variable functions, functions of more than 1 variable, they are what you use for differentiation.
(Wow, I actually remember this stuff from Calculus 3 two years ago!)
Heeeey! Now that stuff doesn’t seem too bad!
Maybe I can be an engineer someday after all. Calc. 3 and Diff.Eq. were the mental brick walls I hit about eight years ago when I was finishing undergrad.
Thanks.
enolancooper writes:
> (Can you tell I don’t know calculus?)
Then why bother to ask this question? If you don’t know anything about calculus, there’s really nothing we can do except point you to a calculus book and tell you to learn some of the terms there before you ask this question again. A couple of people have answered your question in ways that would be useful to someone who knows a reasonable amount about calculus, but without that minimal amount of knowledge there’s no point in trying to explain it to you and no point in your asking the question.
Ah, lighten up Wendel.
I have a soft spot for this topic. In all my formal ejimication the one and only class I ever managed to flunk was a junior level partial differential equations course.
PDQs are used extensively in the aerospace industry. The Wrights probably got along without 'em, but if you want to build anything more complex, you gotta deal with them. They describe the airflow around wings, they describe the stress/strain distribution in structure, and they describe the motion of heat through engines.
The classic example of a PDQ problem is this - suppose you have a metal rod. You put one end in icewater, and stick the other end in a fire. As time goes by, the heat has to find its way through the rod, and so the temperature will vary along the rod. So you have a function of space and time, you need to solve the partial differential equation to describe the temperature at any given point, and any given time.
Mathematically, PDQs can be solved (formally) for only a few simple situations. In practice, for engineering problems, they are solved numerically. There’s a whole bunch of folks out there writing code to deal with aero, thermo, stress, and other problems.
Here is my best non-calculus answer. (though some algebra and geometry is useful)
The derivative can often be graphically described as the slope of an equation.
If you have an equation in one variable, you can draw on a piece of paper. You write the function as y = f(x), draw an XY axis and plot the points. Then you play connect the dots.
It might look like this:
Y axis ^
| /
| _____ /
| / \ /
| / \___/
| /
-------------------->
X axis
The derivative with respect to x, at a particular x is the slope of the curve at that point.
But what happens if you have more than one variable. What if z= f(x,y)? Then you can draw it as a surface. Now it looks like one of those 3-D topological maps.
Let’s pick a point on the side of a “hill” on the topological map. What is the slope there? It depends on which way you go. In the direction of the crest, the slope might be fairly steep. But side to side, the slope might be small or zero.
So, how do you talk about the slope of a surface? You specify the direction in which you are taking the derivative, hence partial derivative. (Only taking derivative in one direction)
The partial derivative of our topological map with respect to x would be the slope in a direction parallel to the x-axis.
It works the same way no matter how many variables you have, it is just harder to draw or visualize.
Which brings up one bonus point for those of you still paying attention to me. The gradient. When a scientist/engineer talks about a gradient, they are talking about the maximum slope at a given point. The gradient axis is the direction of maximum slope.
keeper0: You mention that a gradient is the maximum slope at a given point on a surface, and the gradient axis is the direction of the maximum slope. What is the line called that connects the points along the gradient axis (with the gradient axis changing from point to point)?
If the surface is a mountain, and my goal is to get down as fast as possible, will the best route always be to follow the gradient axis from one point to the next? (I think not. I have an image in my mind where there may be a rapidly descending path for a short while but that it flattens out after a slightly longer while. You’d have been better off not taking the initially steepest slope in order to not to overshoot an even steeper and more sustained slope later on). What is the best path called? Is there an explicit way of determining it for a given surface?
[sub]I do hope these questions makes sense. Please don’t mock me too much if they don’t.[/sub]
The OP was written by someone who knows next to nothing about this. Give REAL LIFE, Readers Digest versions of ACTUAL uses of Partial Diffy Q’s. Like kellymccauley did. All this engineering crap ain’t worth the #2 pencil it is written with if it doesn’t have a PRACTICAL application! Perhaps being a CE made me think of personal interaction with my stuff, but if any of you out there speak English, not slopes of curves, gradient axes, or F(x,y), please speak up. Calculus ain’t shit without application. That’s Real World.
(keeper, nice hill analogy, but a non math/engineering person is making the "blub-lub-lub-lub-lub sound right now)
The shortest path between two points on a particular surface is called the geodesic. For a surface with a well-defined length element ds (trivial examples would be a straight line in the x-direction, where ds = dx, or a circle of radius r in a plane, where ds = rd(theta)*), the length of a path between two points is the integral of ds between those points. The shortest path is thus where this integral evaluates to the smallest answer. One uses methods of the “calculus of variations” to find the maximum and/or minimum of this integral, and thus the geodesic.
More generally there are methods of differential geometry for finding geodesics on curved spaces. An introductory general relativity text should have a decent, practical (as such things go) explanation.
You sure about this?
Some shit in life just isn’t easy to explain properly. Besides, the OP has gone out of their way to post, correctly or otherwise, on a wide range of engineering-related topics here. I’m pretty surprised to find that they don’t know Calculus or what a PDE is. Things that make you go “Hmmm”.