derivative and gradient

shahar said:
What the differences between the two?
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The gradient is a vector of partial derivatives. The derivative is a gradient vector with only one element. So in single-variable functions, you can use them interchangeably, but not the case for multi-variable functions.
 
shahar said:
What the differences between the two?
Click to expand...
It depends on what YOU mean by gradient. This is why you've been asked for your definition.

In some places, "gradient" is used as a synonym for "slope" (and therefore the derivative tells you the slope (of the tangent line)):

Gradient (Slope) of a Straight Line

www.mathsisfun.com

But in other context, it is something different, as others have told you:

math.stackexchange.com

Difference between Slope and Gradient

It has been a few years since studying contour maps. Often I hear slope and gradient interchangeably in describing steepness. Does anyone know any good definitions and analogies of slope and gra...
math.stackexchange.com math.stackexchange.com
 
BigBeachBanana said:
The gradient is a vector of partial derivatives. The derivative is a gradient vector with only one element. So in single-variable functions, you can use them interchangeably, but not the case for multi-variable functions.
Click to expand...
I disagree that you can still use them interchangeably. The gradient will point in a given direction, say either [imath]\hat{i}[/imath] or [imath]-\hat{i}[/imath]. The derivative may be positive or negative but it is just a number. It is a common assumption that a positive or negative number "points" in a positive or negative direction (this is done in Introductory Physics all the time) but this is not quite the same thing. The gradient is a vector whereas the derivative is a scalar.

-Dan
 
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