Consider the partial derivatives
fx(x,y)=4x3y7−20x3y,
fy(x,y)=7x4y6−5x4.
Is there a function f
which has these partial derivatives?
If so, what is it?
f=
Are there any others?
Yes, there is a function f that has these partial derivatives. To find this function, we integrate each partial derivative with respect to its corresponding variable.
First, integrate fx(x,y) with respect to x:
f(x,y) = ∫(4x^3y^7 - 20x^3y) dx
= x^4y^7 - 5x^4y + g(y),
where g(y) is a function of y that arises from the integration constant.
Next, differentiate f(x,y) with respect to y to find fy(x,y):
fy(x,y) = ∂/∂y (x^4y^7 - 5x^4y + g(y))
= 7x^4y^6 - 5x^4 + g'(y).
To have fy(x,y) = 7x^4y^6 - 5x^4, we need g'(y) to be equal to zero. Therefore, we can choose g(y) = C, where C is a constant.
Thus, the function f(x,y) = x^4y^7 - 5x^4y + C satisfies both partial derivatives given.
There may be other functions that have the same partial derivatives, but they would differ by a constant term C.
To determine if there exists a function f with the given partial derivatives, we need to check if the partial derivatives are equal and satisfy the symmetry condition.
Comparing the partial derivatives fx(x, y) and fy(x, y), we see that they are not equal:
fx(x, y) = 4x^3y^7 - 20x^3y
fy(x, y) = 7x^4y^6 - 5x^4
So, the given partial derivatives are not equal.
Therefore, there does not exist a function f with the given partial derivatives.
Hence, f = (does not exist). There are no other functions that satisfy the given partial derivatives.