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I am working independently on getting an intuition about tensors, since they weren't covered adequately in any of my classes. I am working on a simple motivating example I thought of, to see why we need the tensor form of the spring constant instead of the normal scalar form.

Suppose we have a mass connected to two springs that, when at rest, are normal to each other (say one goes along the x-axis and the other the y-axis). Intuitively, it seems that if I move my mass in the y direction it will end up with a force with both a $y$ and $x$ component.

I am working on getting the force at an arbitrary point. I am trying to get this in the form $F=\kappa d$, where $\kappa$ is a tensor and $d$ is the displacement vector, but I am having trouble getting anything that is linear but also contains both x and y components in general.

Can someone help me out to try to get it in this form?

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  • $\begingroup$ It can't be genuinely linear because $\kappa$ will depend on d. But if $F_x= a + b$ and $d=(x,y)$ you could get the form by putting $F_x=\frac{a}{x}x+\frac{b}{y}y$ $\endgroup$
    – Peter
    Jan 10, 2021 at 3:00
  • $\begingroup$ @Peter will it not be linear even in some approximation(e.g we limit x and y to first order)? $\endgroup$
    – JDThinking
    Jan 10, 2021 at 13:26
  • $\begingroup$ It should be possible to do something with partial derivatives. You would get something like $F\approx F_0(x)+ A\delta x$ where $F, x$ are vectors and $A$ is a matrix/tensor. $\endgroup$
    – Peter
    Jan 10, 2021 at 13:36

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