Let \(f: M \rightarrow N\) be a smooth map and \(\omega\) a \(k \cdot\) form on \(N\). Show that \(f^{*}(d \omega)=d\left(f^{*} \omega\right)\).

In the fascinating world of differential geometry, smooth maps are the bridges connecting different shapes and surfaces called manifolds. Imagine two rubber sheets, each one representing a continuous surface without any sharp folds or edges. A smooth map is akin to stretching one rubber sheet to fit the contours of the other perfectly without any tearing or crumpling.

Formally, a smooth map is a function that is infinitely differentiable, meaning you can calculate derivatives of any order without running into any undefined values or abrupt changes. This quality is essential because it allows us to carry properties from one manifold to another through the smooth map, and this transfer is both elegant and structured. When discussing differential forms and their transformations across manifolds, smooth maps play a pivotal role in mapping tangent spaces from one manifold to another, preserving the intricate dance of geometry and calculus.

The concept of the exterior derivative is like having a magical magnifying glass that lets you zoom in on any curve or surface to see its instantaneous rate of change in all possible directions. But instead of examining one variable at a time, it leaps into multi-variable forms all at once.

In technical terms, the exterior derivative takes a differential form of a certain degree (essentially a function that takes in vectors and outputs numbers) and produces another form of one higher degree that represents the infinitesimal circulation around each point. It's the higher-dimensional cousin of the gradient, divergence, and curl from vector calculus. So, when you see the symbol 'd' applied to a form, it's like sprinkling some mathematical fairy dust that reveals deeper insights into the original form's behavior. It's a powerful tool that portrays the inherent structure of manifolds and the functions defined on them, providing a unified language for expressing rates of change.

Pullback of differential forms may sound like a heavyweight move in a cosmic tug-of-war, but it's actually a delicate and precise method of transporting geometric information. Picture a painter copying a landscape onto their canvas, stroke by stroke, capturing every element in precise detail, but laying it down on a different space altogether.

Mathematically, the pullback function takes a differential form defined on one manifold and reinterprets it in terms of another through a smooth map. The resulting form is fundamentally entwined with the original, but it lives on a different arena where it echoes the structure and characteristics of the first manifold. In essence, pulling back forms is all about relaying geometric and physical properties between different stages while respecting their inherent algebraic traits.

Diving deeper into the abstract ocean of differential geometry, tensor calculus is the language that describes how multi-linear maps interact with vectors and points in space. It's the mathematical equivalent of assembling complex LEGO structures, where each brick might represent a different type of force or interaction in the physical world.

Tensors themselves are geometrical entities that generalize the concepts of scalars, vectors, and matrices. When you describe something with a tensor, you're not just saying 'it's this big' or 'it's in this direction'—you're saying 'it's this big in all these different directions and these other ways too.' Tensor calculus is the toolbox you need to work with these incredibly versatile objects, allowing physicists and mathematicians to precisely articulate the relationships between shapes, forces, and motions within the curvy playgrounds of manifolds.

In the grand tapestry of mathematical spaces, manifolds are like patches that make up a quilt. Each patch can lay flat, but when you stitch them together, they can form complex, three-dimensional shapes. This imagery captures the essence of manifolds — they are spaces that locally resemble flat, Euclidean space, but globally can have any number of twists, turns, and loops.

You can imagine a manifold as a shape that can be smoothly unfolded into a piece of paper without tearing or stretching it. This property allows us to apply calculus as if we were working in familiar, flat space, yet acknowledge the shape's true, possibly convoluted form. In the grand scope of differential geometry, manifolds are the stage upon which all other concepts perform — whether it's a smooth map detailing the journey from one surface to another or the elegant dance of differential forms under the operations of the exterior derivative and pullback. Understanding manifolds is essential, as they provide the framework for describing physical phenomena in a geometric context.