Let $M=q(k,t)$ and $N=g(k,t).$ Define $$P=\int Mdk$$ and $$f(t)=N-\frac{\partial P}{\partial t}.$$ Let $$F(t)=\int f(t)dt.$$ Then $$F(t)=\int \left(N-\frac{\partial P}{\partial t}\right)dt.$$ Is it correct to say that $$F(t)=\int N(\text{ taking only those terms that do not contain k})dt?$$ If so, please explain why?
What is $\int f(t)dt$ in this context?
1
$\begingroup$
multivariable-calculus
1 Answers
0
Why do you think this? $N$ is a function of $k$ as you have pointed out, and $P$ does not depend on $N$ in any way? In fact, we can simplify your expression, by noting that:
$$\int \frac{\partial P}{\partial t}\:\mathrm{d}t = P + c$$
So, we can write (using linearity of the integral):
$$F(t) = \int N(k,t)\:\mathrm{d}t - \int M(k,t)\:\mathrm{d}k+c$$
Which doesn't allow us to say anything as strong as what you have put without (at least) more information relating $M(k,t)$ and $N(k,t)$.