ODE now made me think about derivatives and partial derivatives

In summary: If y were independent of x then dy/dx would be zero. This would just mean that x can change without a change in y. But the constraint C = x^2 + y^2 tells you that y is a function of x, at least locally.In summary, the partial derivative of a position function with respect to time would tell you how fast the position changes over time. It would be the rate of change of position with respect to time, assuming y does not change.
  • #1
flyingpig
2,579
1

Homework Statement




Let's say I have a function for a circle

[tex]x^2 + y^2 = C[/tex] where C is a constant.

Then this is a cylinder with the z-axis.

Now in my ODE book, we would normally define it as

[tex]F(x,y) = C = x^2 + y^2[/tex] as a level surface.

Now my question is about what the partial derivative with respect to x mean as opposed to (single-variable calculus) derivative with respect to x mean. Am I losing anything if I take one derivative over the other?

I should mention that many of these problems assume that F(x,y(x)).

[tex]\frac{\partial F}{\partial x} = 2x[/tex]

[tex]\frac{\partial F}{\partial y} = 2y[/tex]

[tex]\frac{\mathrm{d} F}{\mathrm{d} x} = 2x + 2y\frac{\mathrm{d} y}{\mathrm{d} x} = 0[/tex]

So now my question is, what exactly is this
[tex] 2x + 2y\frac{\mathrm{d} y}{\mathrm{d} x} = 0[/tex]
as opposed to
2x
 
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  • #2
[tex]2x+ 2y\frac{dy}{dx}[/tex]
is the rate of change of the function [itex]f(x, y(x))= x^2+ y(x)^2[/itex]
with respect to x- it measures how fast f(x,y(x)) changes as x changes. Of course that will depend upon exactly how y(x) changes as x changes- and that is what dy/dx tells you.

Suppose y(x)= 3x. Then [itex]d(x^2+ y^2)/dx= 2x+ 2y dy/dx= 2x+ 2y(3)= 2x+ 2(3x)(3)= 20x[/itex].That is exactly the same as if you had replaced y with 3x from the start: [itex]x^2+ (3x)^2= x^2+ 9x^2= 10x^2[/itex] so [itex]df/dx= 20x[/tex]
 
  • #3
HallsofIvy said:
[tex]2x+ 2y\frac{dy}{dx}[/tex]
is the rate of change of the function [itex]f(x, y(x))= x^2+ y(x)^2[/itex]
with respect to x- it measures how fast f(x,y(x)) changes as x changes.

I thought that was what the partial derivative with respect to x is
 
  • #4
No, the partial derivative of f with respect to x is the rate of change as x change assuming y does not change.
 
  • #5
flyingpig said:
I should mention that many of these problems assume that F(x,y(x)).

Right.

given a small change in x, dx, the change is x^2 is close to 2xdx and the change in y^2 is close to 2ydy. But dy = (dy/dx)dx.

If y were independent of x then dy/dx would be zero. This would just mean that x can change without a change in y. But the constraint C = x^2 + y^2 tells you that y is a function of x, at least locally.
 
  • #6
lavinia said:
Right.

given a small change in x, dx, the change is x^2 is close to 2xdx and the change in y^2 is close to 2ydy. But dy = (dy/dx)dx.

If y were independent of x then dy/dx would be zero. This would just mean that x can change without a change in y. But the constraint C = x^2 + y^2 tells you that y is a function of x, at least locally.

If I were to graph all three of those "derivatives" what would they look like? How do yuo even graph Fx alone?
 
  • #7
Actualyl could I get a unit representation?

Say F(x,y) was a position function of time (perhaps x(t) = x and y = y(t))

what would the different derivatives tell me?
 

1. What are Ordinary Differential Equations (ODEs)?

ODEs are mathematical equations that describe the relationship between a function and its derivatives. They are commonly used in physics, engineering, and other scientific fields to model dynamic systems.

2. What is the difference between a derivative and a partial derivative?

A derivative is the rate of change of a single variable with respect to another variable. A partial derivative, on the other hand, is the rate of change of a multivariable function with respect to one of its variables while holding the others constant. In other words, a partial derivative considers only changes in one variable while keeping the others fixed.

3. How are ODEs used in scientific research?

ODEs are used in scientific research to model and analyze various physical phenomena, such as the motion of objects, chemical reactions, and population dynamics. ODEs allow scientists to make predictions and understand the behavior of these systems over time.

4. What are some methods for solving ODEs?

There are several methods for solving ODEs, including numerical methods like Euler's method and analytical methods like separation of variables. The best method to use depends on the complexity of the ODE and the desired level of accuracy in the solution.

5. How do derivatives and partial derivatives relate to ODEs?

Derivatives and partial derivatives are essential tools in solving ODEs. In fact, ODEs are defined in terms of derivatives, and partial derivatives are used to solve systems of ODEs. They allow scientists to understand how a system changes over time and make predictions about its behavior.

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