Let’s start with a differentiable manifold M of dimension . Throughout this topic, we denote by a point on and its local chart (at ). A point is determined by hence it is often identified with . We usually denote by the local coordinates of .

**Definition 1**. A tangent vector at is a map defined on the set of the differentiable functions in a neighborhood of , where satisfies the following conditions

- is linear, that is to say: if , then .
- if is flat at , i.e. at .
- .

**Definition 2**. The tangent space at is the set of tangent vectors at .

From the definition 1, let us show that the tangent space of definition 2 has a natural vector space structure of dimension . We set

and .

With this sum and this product, is a vector space. And now let us exhibit a basis. It is reasonable to define the tangent vector at . Precisely,

**Definition 3**. The tangent vector at is defined to be

.

The vectors are independent and they form a basis for . We usually call the directional derivative of in the direction . For an arbitrary vector , one can define the directional derivative of in the direction as following

where denotes the -th component of in this coordinate chart.

We now assume further that is a Riemannian manifold where is its metric. We are now in a position to define gradient for a smooth function.

**Definition 4**. For any smooth function on a Riemannian manifold , the gradient of is the vector field such that for any vector field ,

, i.e.

where denotes the inner product of tangent vectors at defined by the metric* *.

We now express the local form of the gradient at . By definition 4, one has in the local coordinates

.

If we assume we then have

.

Thus

which implies after multiplying both sides by the matrix

.

Therefore,

.

We end this topic by showing what is? Roughly speaking, at a point since is a vector, then is nothing but its magnitude. To be exact, one defines

.

In Riemannian geometry, the lower index means differentiation and the upper index means component, therefore, we usually use to denote the quantity . With this notation, .

of

in the direction

* *
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