# Dimensional analysis seems wrong for this equation: z = -1/(x^2+y^2)

• I
dyn
Hi
I've just done a question regarding a marble moving on a surface given by z = -1/(x2+y2)
In this case what happens with dimensional analysis ? x and y have dimensions of length while z has dimensions of 1/(length)2.
Is this question badly written or do i just accept that i won't be able to perform dimensional analysis on any later terms that are produced ?
Thanks

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Homework Helper
Gold Member
So if z is a distance, and x and y are distances, then there needs to be some implied dimension in the 1 in the numerator, to make the dimension of z be correct.

It's kind of like you can have y = x^2 mathematically, but if it represents something physical, then you might have to multiply by a coefficient (could be 1) with some dimension (or units).

• PeroK and berkeman
Gold Member
2022 Award
HI
I've just done a question regarding a marble moving on a surface given by z = -1/(x2+y2)
In this case what happens with dimensional analysis ? x and y have dimensions of length while z has dimensions of 1/(length)2.
Is this question badly written or do i just accept that i won't be able to perform dimensional analysis on any later terms that are produced ?
Thanks
The question is badly formulated. There's some constant with the appropriate dimension missing. If it's in a math textbook it's simply that mathematicians nowadays don't care for correct physics. If it's a physics textbook it's simply a bad book ;-)).

• dyn and scottdave
2022 Award
Is this question badly written or do i just accept that i won't be able to perform dimensional analysis on any later terms that are produced ?
It's a badly written question. But you can argue that the real surface is ##z'=Az##, where ##A## has dimensions of length cubed and equals one in whatever units you are using. At any point in your working you can simply substitute ##z'/A## for ##z## and check dimensions.

• dyn
ergospherical
just an opinion but I think ##z=-1/(x^2+y^2)## is better written than ##z = -(1\ \mathrm{m^3})/(x^2 + y^2)## or ##(z/\mathrm{m}) = -1/((x/\mathrm{m})^2 + (y/\mathrm{m})^2)## or whatever because the emphasis is clearly on studying the properties of the surface and not confusing the student with weird dimensional factors

• Office_Shredder
Gold Member
2022 Award
I'd simply write ##z=-A/(x^2+y^2)##, where ##A=\text{const}##. Of course, ##A## has the dimension ##\text{length}^3##. A dimensionally wrong equation is a nogo in any physics book!

• dyn and scottdave
dyn
The question is badly formulated. There's some constant with the appropriate dimension missing. If it's in a math textbook it's simply that mathematicians nowadays don't care for correct physics. If it's a physics textbook it's simply a bad book ;-)).
It was on a university maths exam paper !

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