Exploring the Meaning Behind $\nabla^2 f(x) = -4$

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In summary, the first problem involves an integral with a function f that depends on the absolute value of xi and the partial derivatives of a function phi with respect to xi and eta. The result of this integral is 0, which is apparently obvious after only doing the eta integral. In the second problem, we are asked to find the Laplacian of a function f(x)=-|x|^2, with conflicting answers of -4 and -6. The discrepancy arises from the number of dimensions of x, with the Laplacian being 2 for two dimensions and 6 for three dimensions.
  • #1
latentcorpse
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1, apparently

[itex]\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(-|\xi|) 4 \phi_{\xi \eta} d \xi d \eta=0[/itex]

also apparently this is obvious after only doing the eta integral. any ideas why?

2, what is [itex]\nabla^2 f(x)[/itex] where [itex]f(x)=-|x|^2[/itex]
the answers say its -4
when i just do the second derivative with respect to x i get -2
and when i use index notation i get -6 since
[itex]\partial_i \partial_i r_j r_j = \partial_i (2 \delta_{ij} r_j)=2 \partial_i r_i = 2 \delta_{ii} =6[/itex]

what is going on here?
 
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  • #2
Latex still isn't working so I had to look at your LaTex code.

For problem 1, what is "phixi, eta"?

For the second, "nabla^2" is usually defined for functions of several variables. If f(x) really is |x|2= x2, then the second derivative is 2 as you say. If x is a two dimensional vector with x= <x, y>, then |x|= x2+ y2 and its Laplacian is 4. If x is a three dimensional vector with x= <x, y, z> then |x|= x2+ y2 and its Laplacian is 6. (More generally, if x is an n-dimensional vector, its Laplacian is 2n.)
 
  • #3
ok. thanks i follow the laplacian thing now.

phi_xi,eta is the derivative of phi wrt xi and then wrt eta
 

1. What does $\nabla^2 f(x) = -4$ mean?

The symbol $\nabla^2$ represents the Laplace operator, which is a mathematical operator used to measure the curvature of a function. In this context, it is being applied to the function f(x). The equation as a whole is stating that the curvature of f(x) is equal to -4.

2. How is $\nabla^2 f(x) = -4$ used in science?

This equation is commonly used in physics and engineering, specifically in the fields of fluid dynamics and electromagnetism. It is used to describe the behavior of various physical phenomena, such as heat transfer and electromagnetic fields.

3. What does $\nabla^2 f(x) = -4$ tell us about the function f(x)?

Since the equation is describing the curvature of f(x), it tells us that the function has a constant curvature of -4. This means that the function is either concave or convex, depending on the sign of the curvature.

4. How is $\nabla^2 f(x) = -4$ related to the gradient of f(x)?

The gradient of a function is a vector that points in the direction of steepest increase of the function. The Laplace operator, represented by $\nabla^2$, is used to measure the change in the gradient. In this case, the equation is stating that the change in the gradient of f(x) is equal to -4.

5. Can $\nabla^2 f(x) = -4$ have multiple solutions?

Yes, there can be multiple functions that satisfy the equation $\nabla^2 f(x) = -4$. These functions would have a constant curvature of -4, but could have different shapes and values at different points. This is similar to how there can be multiple solutions to an equation like x = 4.

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