Trouble with function definition

[monea83]

Given is the following function (nevermind what the function h is):

$$g(t, q) = \int_0^1 \frac{\partial h(ts, q)}{\partial(ts)} ds$$

This function is supposed to be defined for t = 0. However, I don't see how - the partial derivative in the integral then becomes $$\frac{\partial h(0, q)}{\partial(0)}$$ and this does not make any sense to me.

If it's any help, this was taken from "do Carmo, Riemannian Geometry", Chapter 0, Lemma 5.5

 

[mathman]

Any derivative is a function, which can take on a particular value for a particular value of the argument. For your integral, by setting t=0, the integrand is no longer dependent on s, so the integral is simply the value of the partial derivative at t=0.

 

[monea83]

Thanks for your answer, I think I understand it better now. The one thing that still bothers me is the $$\partial(ts)$$. How is this to be interpreted? Is it just a placeholder that says "partial differentiation by the first argument"?

 

[mathman]

That's what it looks like to me. Let u=ts, take the partial derivative with respect to u and then set u=ts after-wards before integrating with respect to s.