Finding Functions: v(x,y) with v_x & v_y

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In summary, to find all functions v(x,y) such that both v_x(x,y) = x^2 + y and v_y(x,y) = x - y^3, we can use the method of finding a potential function for a conservative vector field. This involves integrating the partial derivatives of v with respect to x and y, and then determining the constant functions by differentiating again with respect to x and y. This method is more reliable than simply guessing a solution, and it can be applied to many problems involving partial derivatives.
  • #1
Trung
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Homework Statement



Find all functions v(x,y) such that both v_x(x,y) = x^2 + y and v_y(x,y) = x - y^3.

Homework Equations





The Attempt at a Solution



I have no idea how to start. My instructor says to guess the solution, which I have tried, but failed.
 
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  • #2
Integrate V_x with respect to x, then you have V(x,y) = x^3/3 + xy + f(y), where f is an arbitrary function. Then differentiate that expression with respect to y and compare to V_y.
 
  • #3
Trung said:
I have no idea how to start. My instructor says to guess the solution, which I have tried, but failed.
He gave you THAT advice? I really don't know what to make of it, but guessing definitely doesn't work for me.

As nicksauce said, the best way to do this is to treat to find F(x,y) such that its partial derivatives wrt x,y are given by your question. Have you learned what are conservative vector fields/functions yet? The method we're using here is the same as that used to find the potential function, given a conservative vector field.
 
  • #4
Trung said:

Homework Statement



Find all functions v(x,y) such that both v_x(x,y) = x^2 + y and v_y(x,y) = x - y^3.

Homework Equations





The Attempt at a Solution



I have no idea how to start. My instructor says to guess the solution, which I have tried, but failed.
"Guessing a solution" (and then checking) is a good way to solve a problem if you have lots of experience!

Notice, by the way, that if vx= x2+ y then, differentiating again with respect to y, vyx= 1. If vy= x- y3 then, differentiating again with respect to x, vxy= 1. The fact that those are the same (so that the "mixed" derivatives are the same) tells us we can find such a v!

Follow nicksauce's advice.
 
  • #5
It may help to look at the problem in differential form:



(This differential formula can be though of as the definition of the partial derivatives!)


then integration yields:


(Trust the notation!)

Note that when you integrate with respect to say your constant of integration is only "constant" with respect to and so can be any function of alone. (Since x and y are both independent variables you treat y as a constant when integrating with respect to x and vise versa).

Say you do the above integration and get:



(You can absorb the left hand side constant into the other "constant" functions.)
You'll know F and G but must figure out and . You can do this by differentiating again once with each of x or y.
 
  • #6
Oops! I erred Terribly:
I should have said that integration yields:



you would then get something like:

 
Last edited:

FAQ: Finding Functions: v(x,y) with v_x & v_y

What is the purpose of finding functions with v(x,y) and v_x & v_y?

The purpose of finding functions with v(x,y) and v_x & v_y is to understand the relationship between two variables, x and y, and their corresponding velocities, v_x and v_y. This allows for a better understanding of how changes in one variable affect the other, and vice versa.

How do you find a function with v(x,y) and v_x & v_y?

To find a function with v(x,y) and v_x & v_y, you need to have a set of data or equations that describe the relationship between the variables. From there, you can use mathematical techniques such as integration or differentiation to find the function that best fits the data or equations.

What are some common applications of finding functions with v(x,y) and v_x & v_y?

Some common applications of finding functions with v(x,y) and v_x & v_y include studying the motion of objects in physics, analyzing fluid dynamics in engineering, and predicting changes in population growth in biology.

How can finding functions with v(x,y) and v_x & v_y be useful in real-world scenarios?

Finding functions with v(x,y) and v_x & v_y can be useful in real-world scenarios by providing a mathematical model to predict and analyze the behavior of systems and variables. This can aid in decision making, problem solving, and understanding complex phenomena.

Are there any limitations to finding functions with v(x,y) and v_x & v_y?

One limitation of finding functions with v(x,y) and v_x & v_y is that it assumes a linear relationship between the variables and their velocities. In reality, this relationship may be more complex and may require more advanced mathematical techniques to accurately model. Additionally, finding functions with v(x,y) and v_x & v_y may not account for external factors that can affect the variables and their velocities.

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