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I cannot give any reasons for the purpose of connections, as I myself have yet to study them. As for differential forms, I will give you a topological reason to study them. The topological motivation is de Rham cohomology. An inherently analytic object such as a differential $k$-form $\omega$, depending on whether or not it solves the equation $\omega = d\eta$, can help to determine the de Rham cohomology groups$$H^k_{dR}(M) := \frac {\ker(d:\Omega^k(M)\to\Omega^{k+1}(M))}{\operatorname{im} (d:\Omega^{k-1}(M)\to\Omega^k(M))}$$ which, as it turns out, are homotopy invariants (and by a theorem of de Rham himself, isomorphic to the singular cohomology with real coefficients $H^k(M;\mathbb R)$). Thus, by studying the solvability of certain differential equations involving forms, we can determine something about the topology of the underlying topological space. $$\\\\$$As for the Lie derivative, the real motivation is that we want to do differential calculus with vector/tensor fields. So, we take the flow of the vector field whose direction we want to differentiate in, pull back the vector field being differentiated, and take the derivative at $t=0$, to explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. What it allows us to do is to "transport" a vector field at one point to a vector at another, and find the rate of change of that transformation. Why do we do this? Because, it is a step along the way in our quest of generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection/covariant derivative (My source for this is the following Math.SE question: Link). As for your question on the tangent bundle, it allows us to study things globally, and we can use the machinery of connections, parallel transport, etc. Hope this helps. $$\\\\$$ My main source is John Lee's Introduction to Smooth Manifolds.

The real motivation for the Lie derivative is doing differential calculus with vector fields. If we want fo differentiate the vector field $W$ in the direction of the vector field $V$, we take the flow of $V$ through time, use it to pull back $W$, and take the derivative at $t=0$.

To explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. It allows us to transport a vector field at one point to a vector at another, and find the rate of change of that transformation. 

Why do we do this? Because it is a step towards generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. 

However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection.

(My source for the last comment is the following Math.SE question: Link. My main source generally is John Lee's Introduction to Smooth Manifolds.)

I cannot give any reasons for the purpose of connections, as I myself have yet to study them. As for differential forms, I will give you a topological reason to study them as well as a geometric reason to study them. The topological motivation is de Rham cohomology. An inherently analytic object such as a differential $k$-form $\omega$, depending on whether or not it solves the equation $\omega = d\eta$, can help to determine the de Rham cohomology groups$$H^k_{dR}(M) := \frac {\ker(d:\Omega^k(M)\to\Omega^{k+1}(M))}{\operatorname{im} (d:\Omega^{k-1}(M)\to\Omega^k(M))}$$ which, as it turns out, are homotopy invariants (and by a theorem of de Rham himself, isomorphic to the singular cohomology with real coefficients $H^k(M;\mathbb R)$). Thus, by studying the solvability of certain differential equations involving forms, we can determine something about the topology of the underlying topological space. $$\\\\$$As for the Lie derivative, the real motivation is that we want to do differential calculus with vector/tensor fields. So, we take the flow of the vector field whose direction we want to differentiate in, pull back the vector field being differentiated, and take the derivative at $t=0$, to explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. What it allows us to do is to "transport" a vector field at one point to a vector at another, and find the rate of change of that transformation. Why do we do this? Because, it is a step along the way in our quest of generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection/covariant derivative (My source for this is the following Math.SE question: Link). As for your question on the tangent bundle, it allows us to study things globally, and we can use the machinery of connections, parallel transport, etc. Hope this helps. $$\\\\$$ My main source is John Lee's Introduction to Smooth Manifolds.

I cannot give any reasons for the purpose of connections, as I myself have yet to study them. As for differential forms, I will give you a topological reason to study them. The topological motivation is de Rham cohomology. An inherently analytic object such as a differential $k$-form $\omega$, depending on whether or not it solves the equation $\omega = d\eta$, can help to determine the de Rham cohomology groups$$H^k_{dR}(M) := \frac {\ker(d:\Omega^k(M)\to\Omega^{k+1}(M))}{\operatorname{im} (d:\Omega^{k-1}(M)\to\Omega^k(M))}$$ which, as it turns out, are homotopy invariants (and by a theorem of de Rham himself, isomorphic to the singular cohomology with real coefficients $H^k(M;\mathbb R)$). Thus, by studying the solvability of certain differential equations involving forms, we can determine something about the topology of the underlying topological space. $$\\\\$$As for the Lie derivative, the real motivation is that we want to do differential calculus with vector/tensor fields. So, we take the flow of the vector field whose direction we want to differentiate in, pull back the vector field being differentiated, and take the derivative at $t=0$, to explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. What it allows us to do is to "transport" a vector field at one point to a vector at another, and find the rate of change of that transformation. Why do we do this? Because, it is a step along the way in our quest of generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection/covariant derivative (My source for this is the following Math.SE question: Link). As for your question on the tangent bundle, it allows us to study things globally, and we can use the machinery of connections, parallel transport, etc. Hope this helps. $$\\\\$$ My main source is John Lee's Introduction to Smooth Manifolds.

I cannot give any reasons for the purpose of connections, as I myself have yet to study them. As for differential forms, I will give you a topological reason to study them as well as a geometric reason to study them. The topological motivation is de Rham cohomology. An inherently analytic object such as a differential $k$-form $\omega$, depending on whether or not it solves the equation $\omega = d\eta$, can help to determine the de Rham cohomology groups$$H^k_{dR}(M) := \frac {\ker(d:\Omega^k(M)\to\Omega^{k+1}(M))}{\operatorname{im} (d:\Omega^{k-1}(M)\to\Omega^k(M))}$$ which, as it turns out, are homotopy invariants (and by a theorem of de Rham himself, isomorphic to the singular cohomology with real coefficients $H^k(M;\mathbb R)$). Thus, by studying the solvability of certain differential equations involving forms, we can determine something about the topology of the underlying topological space. $$\\\\$$As for the Lie derivative, the real motivation is that we want to do differential calculus with vector/tensor fields. So, we take the flow of the vector field whose direction we want to differentiate in, pull back the vector field being differentiated, and take the derivative at $t=0$, to explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. What it allows us to do is to "transport" a vector field at one point to a vector at another, and find the rate of change of that transformation. Why do we do this? Because, it is a step along the way in our quest of generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection/covariant derivative (My source for this is the following Math.SE question: Link). As for your question on the tangent bundle, it allows to study things globally, and we can use the machinery of connections, parallel transport, etc. Hope this helps. $$\\\\$$ My main source is John Lee's Introduction to Smooth Manifolds.

I cannot give any reasons for the purpose of connections, as I myself have yet to study them. As for differential forms, I will give you a topological reason to study them as well as a geometric reason to study them. The topological motivation is de Rham cohomology. An inherently analytic object such as a differential $k$-form $\omega$, depending on whether or not it solves the equation $\omega = d\eta$, can help to determine the de Rham cohomology groups$$H^k_{dR}(M) := \frac {\ker(d:\Omega^k(M)\to\Omega^{k+1}(M))}{\operatorname{im} (d:\Omega^{k-1}(M)\to\Omega^k(M))}$$ which, as it turns out, are homotopy invariants (and by a theorem of de Rham himself, isomorphic to the singular cohomology with real coefficients $H^k(M;\mathbb R)$). Thus, by studying the solvability of certain differential equations involving forms, we can determine something about the topology of the underlying topological space. $$\\\\$$As for the Lie derivative, the real motivation is that we want to do differential calculus with vector/tensor fields. So, we take the flow of the vector field whose direction we want to differentiate in, pull back the vector field being differentiated, and take the derivative at $t=0$, to explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. What it allows us to do is to "transport" a vector field at one point to a vector at another, and find the rate of change of that transformation. Why do we do this? Because, it is a step along the way in our quest of generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection/covariant derivative (My source for this is the following Math.SE question: Link). As for your question on the tangent bundle, it allows us to study things globally, and we can use the machinery of connections, parallel transport, etc. Hope this helps. $$\\\\$$ My main source is John Lee's Introduction to Smooth Manifolds.