When you look at a solution for a problem involving, say, the volume of a parallelepiped in Apostol, you don't just see a formula applied. You see a logical argument that treats the volume integral as a determinant of a linear transformation. The solution manual reads less like a "cheat sheet" and more like a formal logic paper.
Several mathematicians and students have started open-source projects to typeset every solution in LaTeX. These are often the most accurate and readable options. tom m apostol calculus volume 2 solution manual