- #1

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- TL;DR Summary
- Confusion about the gradient term in the derivation.

**Definition**: Let

*f*be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P##

*be a tangent vector*to it. Then the following number is the derivative of a function w.r.t. the tangent vector

$$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t \mathbf{v}) \big)|_{t=0}$$

Then there is this

**Lemma**

*:*If ##\mathbf{v}_p= (v_1,v_2,v_3)_P## is a tangent vector to ##\mathbf{R}^3##

*, then*

[tex]\mathbf{v}_p[\mathit{f}]= \sum_i v_i \frac{d \mathit{f}}{dx_i}(P)[/tex]

**1)**How does this gradient term come into the picture? Since, [itex]\mathit{f}(\mathbf{P}+ t \mathbf{v})[/itex], do we write the argument as [itex]\mathbf{x}(t)[/itex] and then apply the chain-rule in the

**definition**?

**2)**Could this be thought of as a dot-product ##\mathbf{v}_P \,. \, \nabla \mathit{f}|_p ##?

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