An example I was thinking of is area between two curves in Calc I/II.
in Multivariable Calculus this was the equivalent
$$\int_{a}^{b} \int_{f(x)}^{g(x)} dxdy = \int_{a}^{b} (g(x)-f(x)) dx$$
so I got to thinking can this work in other topics? namely the volume topics in Calc I/II: Volumes of revolution, Cross-Section Volume, etc.
On
The Fundamental Theorem of Calculus, the most important notion in Calc I/II, certainly has a very nice multivariable equivalent: the Stokes theorem. The FTC gives you the integral of $f$ along an interval in terms of the values of $F$ at the boundary, very much like the vector calc form of Stokes:
$$\iint_\Sigma \nabla \times \mathbf{F} \cdot d\boldsymbol{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot d\mathbf{r}$$
In its modern version, Stokes' theorem generalizes this to orientable manifolds of say dimension $n$ by looking at the antiderivative at the boundary, which would be $n-1$-dimensional.
It's difficult to decide what "every topic" entails. However, it's true that the calculus on $\mathbb{R}$ which you studied in the first two courses is a subset of calculus on $\mathbb{R}^n$, which is itself a subset of the large field known as analysis.