^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.6 - Functions on M|2.6 Functions on M]] | [[Schutz - 2.8 - Basis Vectors and Basis Vector Fields|2.8 Basis Vectors and Basis Vector Fields]]>> # Vectors and Vector Fields Consider a [[Schutz - 2.5 - Curves#^5c64cb|curve]] passing through a point $P \in M$ for some manifold $M$, described by the equations $x^i = x^i(\lambda)$, where $\lambda$ parametrizes the curve. Moreover, consider a [[Schutz - 2.6 - Functions on M#^6ded65|differentiable function on M]] $f(x^i)$. At each point of the curve, $f$ has a value. Therefore, along the curve, there is a differentiable functoin $g(\lambda)$ which gives the value of $f$ at the point whose parameter value is $\lambda$: $ g(\lambda) = f(x^i(\lambda)) $ Differentiating and using the chain rule gives: $ \frac{dg}{d\lambda} = \sum_i \frac{dx^i}{d\lambda} \frac{\partial f}{\partial x^i} = \frac{dx^i}{d\lambda} \partial_i f $ Since this is true for any function $g$, we can write: $ \frac{d}{d\lambda} = \sum_i \frac{dx^i}{d\lambda} \frac{\partial}{\partial x_i} $ In the ordinary view of Euclidean vectors, one can say that the set of numbers $\{\frac{dx^i}{d\lambda}\}$ are components of the vector tangent to $x^i(\lambda)$. The $dx^i$ are displacements and the $d\lambda$ simply change the scale. Since each curve has a [[Schutz - 2.5 - Curves#^6a34c8|unique parameter]], there is a unique set $\{\frac{dx^i}{d\lambda}\}$ which are said to be the components of *the* tangent to the curve. Thus, by our definition of a parameterized differential curve, each curve has a *unique* tangent vector. Every vector is tangent to an infinite number of curves through $P$ as shown in the figure below. ![[Pasted image 20210123200206.png]] Since manifolds have no notion of distance, we need to define vectors more abstractly, without such a notion (i.e. not regard $dx^i$ as a displacement). In our definition, we should rely only on infinitesimal neighborhoods of points of $M$. Suppose $a, b \in \mathbb{R}$ and $x^i = x^i(\mu)$ is another curve through $P$. Then at $P$ we have: $ \frac{d}{d\mu} = \sum_i \frac{dx^i}{d\mu} \frac{\partial}{\partial x_i} $ and $ a \frac{d}{d\lambda} + b \frac{d}{d\mu} = \sum_i \left( a \frac{dx^i}{d\lambda} + b \frac{dx^i}{d\mu}\right) \frac{\partial}{\partial x_i} $ Now, the numbers $\left( a \frac{dx^i}{d\lambda} + b \frac{dx^i}{d\mu}\right)$ are the components of a new vector which is tangent to *some* curve at the point $P$. Thus, there must exist a curve with some parameter $\phi$ for which: $ \frac{d}{d\phi}= \sum_i \left( a \frac{dx^i}{d\lambda} + b \frac{dx^i}{d\mu}\right) \frac{\partial}{\partial x_i} $ and we get: $ \frac{d}{d\phi} = a \frac{d}{d\lambda} + b \frac{d}{d\mu} $ We can conclude that: *the directional derivatives along curves form a vector space at P*. We have shown that they obey closure under linear combinations above, and the remaining axioms of vector spaces are trivial to show. > **Tangent Vector** > The tangent vector to a curve parametrized by $\lambda$ on a manifold $M$ is: > $ > \frac{d}{d\lambda} = \sum_i \frac{dx^i}{d\lambda} \frac{\partial}{\partial x_i} > $ > From the equation above it follows that $\{\partial/\partial x^i\}$ defines a basis for this vector space, and the components of $d/d\lambda$ are $\{dx^i/d\lambda\}$ on this basis. ^ab2ebc This view of vectors has 3 advantages: 1. It does not involve displacements over finite separations 2. It makes no mention of coordinates, i.e. you do not need to specify how a vector transforms as 3. The derivative is a kind of motion along the curve, which is what conceptually what a tangent vector generates. This couples analysis and geometry with very powerful consequences. Note that now, only vectors at the same point $P$ can be added together. Vectors at two different points have no relation with one another. > **Tangent Space** > The tangent space to $M$ at the point $P$, called $T_P$, is the space in which tangent vectors lie. It has the same dimension $n$ as the manifold. ^4e794a > **Vector Field** > A rule for assigning a vector at each point of $M$. ^a9b237