Smoothness measure in thin plate splines


Why do thin splines define the following as a smoothness measure
(subscripts indicate derivatives)

(f_xx)^2 + (f_xy)^2 + (f_yy)^2?

Can someone help me to understand what is the motivation behind 2nd
order derivatives. For example if one used

(f_x) ^ 2 + (f_y) ^ 2,

then this term will penalize high gradient values too, so will yield
smoother f functions. Why do we need 2nd order derivatives?

I tried to find this on Google, with no avail.

Thank you


.

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