What is the Gradient of a Vertical Line?

[Mathpower]

Homework Statement

Basic question (i know!):
I cannot remember whether the gradient of a straight vertical line is infinite or undefined.
Eg: what is the gradient of the line x=3

The Attempt at a Solution

I know it is either undefined or infinite. Can't remember which one!
PLEASE ANSWER...I NEED TO USE THIS IN MY CALCULATIONS FOR A DIFFERENTIATION QUESTION!

 

[Sourabh N]

Have you looked at the wiki article for Gradient?

 

[Mathpower]

That article definitely doesn't help! Thank you for actually replying though...

It seems to me that you other guys do not know the answer either! LOL...

 

[Sourabh N]

It doesn't? Well, first off, Gradient is an operator. It acts on a scalar or a vector function (I believe it can be generalized to tensors, too).
Since you say "straight vertical line", I will assume you are working in the 2 dimensional flat space. Since you use the Cartesian coordinates, well, what is the Gradient operator in flat 2 dimensional Cartesian coordinates?

 

[Mathpower]

Thank you for coming back. I do know what a gradient is!
Yes I am working with a 2 dimensional plane, not a 3 dimensional one!
Yes...cartesian plane.
The operator is positive.
Ok...for the line y=7 the gradient is 0. For the line y=x^3 the gradient at any point can be denoted by the function =3x^2. But what is the gradient of a vertical line...
Too make things simple: just tell me what the gradient of the following equation is (please):
x=3

 

[Sourabh N]

Ah, I see what you mean.
Gradient of a straight vertical line is not defined. You can argue that out by calculating the limit of the slope of a line slightly off the vertical line, on both sides.

 

[Mathpower]

oh...thank you so much.
But I have to enquire:
Is the gradient of the graph infinite (if your answer is no...is it infinite atleast from a technical stand point of view).

Once again thank you very much...I hope I am not wasting your time.

Edit: wait a minute...isn't infinity undefined anyway?...hence the reason why the gradient is called undefined?

 

[Sourabh N]

No problem :)

As I said in the previous post, calculate the gradient of the vertical line by taking the limit of gradient of a line infinitesimally titled from the vertical line. There will be two such lines, one titled on one side of the vertical line, other titled to the other side. The limits of the two cases will not agree, thus making the gradient "undefined" for a vertical line.

 

[Mathpower]

Awesome...thank you for all your help.
I hope you get rewarded for helping.