once exposed to linear algebra and vector calculus, build calculus on manifolds along many examples i.e. go from real $\mathbb{R}$-abstract multilinear till the de Rham complex illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology riemannian geometry, ect

Once students have been exposed to linear algebra and vector calculus, build calculus on manifolds using many examples; i.e. go from real $\mathbb{R}$-abstract multilinear to the de Rham complex, illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology Riemannian geometry, etc.

once exposed to linear algebra and vector calculus, build calculus on manifolds along many examples i.e. go from real $mathbb{R}$-abstract multilinear till the de Rham complex illustrating in $mathbb{R}^3$. All that easen differential geometry, differential topology riemannian geometry, ect

once exposed to linear algebra and vector calculus, build calculus on manifolds along many examples i.e. go from real $\mathbb{R}$-abstract multilinear till the de Rham complex illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology riemannian geometry, ect