It doesn't generally hold true for all functions. The sample function above holds the property because of the nature behind cos and sin functions, the second derivative of them are an constant multiply of themselves. i.e. There are solutions for the differential equations f'' + kf = 0.
dranzer
Yup the laplacian would not be a constant of the function for polynomial functions
keenan
@FeiFeiFei Yep, exactly! We picked a really special function here, which happens to be an "eigenfunction" of the Laplacian. In other words, $\Delta f = \lambda f$ for some constant $\lambda$ (in this case, $\lambda = -9$). This eigenvalue/eigenvector relationship is very much like the usual relationship for matrices.
It doesn't generally hold true for all functions. The sample function above holds the property because of the nature behind cos and sin functions, the second derivative of them are an constant multiply of themselves. i.e. There are solutions for the differential equations f'' + kf = 0.
Yup the laplacian would not be a constant of the function for polynomial functions
@FeiFeiFei Yep, exactly! We picked a really special function here, which happens to be an "eigenfunction" of the Laplacian. In other words, $\Delta f = \lambda f$ for some constant $\lambda$ (in this case, $\lambda = -9$). This eigenvalue/eigenvector relationship is very much like the usual relationship for matrices.