I'm going through a basic introduction to tensors, specifically https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf and I'm confused by the author when he defines vectors as directional derivatives at the bottom of page 3.(adsbygoogle = window.adsbygoogle || []).push({});

He defines a simple example in which

[tex] ƒ(x^j) = x^1 [/tex]

and then goes on to write the directional derivative along a vector [itex] v[/itex] as:

[tex] v ⋅ ∇ ƒ(x^j) = v ⋅ ∇ x^1 = v^i ⋅ \delta^1_i = v^1 [/tex]

Next the author says that all we need to do to get the vector out is to feed the corresponding component into the directional derivative

[tex] v^i = v ⋅ ∇ x^i [/tex]

I don't understand what is being said here at all. This formula only works for the first component and only when the function is is [itex] f(x^j) = x^1 [/itex]. If we wanted to get the second component out of the vector we would end up with 0 no matter what the vector actually was. If our function was [itex] f(x^j) = (x^1)^2 [/itex] then we would have

[tex] v ⋅ ∇f(x^j) = v ⋅ (2x^1, 0, 0) [/tex]

and the first component of our vector would depend on the point of evaluation.

Directional derivatives are scalars, I don't understand how you could equate them with vectors. Directional derivatives need context (a function, a point of evaluation, and a direction from that point) but a vector alone needs none of that.

I'm very confused by this section and I think i completely mis-interpret what is trying to be said. What is meant by all this?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I How is a vector a directional derivative?

Tags:

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for vector directional derivative | Date |
---|---|

I Geometric meaning of complex null vector in Newman-Penrose | Yesterday at 1:34 PM |

I Locality of a vector space | Apr 15, 2018 |

Simple question on future-directed vectors | Jan 4, 2016 |

Is time a vector? if it is, what is its direction? | Jul 6, 2015 |

Confused with directional derivative definition of vectors | Jul 31, 2012 |

**Physics Forums - The Fusion of Science and Community**