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Gradient of a function is $\langle f_x(x,y),f_y(x,y) \rangle$. But I don't understand this gradient vector shows what. When I find gradient of some function, that vectors represents what? Thank you.

Ayca Ekin
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  • The following sources might be useful: a)http://en.wikipedia.org/wiki/Gradient, b)http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/, c)https://www.youtube.com/watch?v=sMcxbJWwlZo – nullgeppetto May 18 '15 at 09:02

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The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that

  1. Points in the direction of greatest increase of a function.
  2. Is zero at a local maximum or local minimum (because there is no single direction of increase).

The term “gradient” is typically used for functions with several inputs and a single output (a scalar field).

We know the definition of the gradient: a derivative for each variable of a function. enter image description here

Suppose,

enter image description here

enter image description here

If we want to find the direction to move to increase our function the fastest, we plug in our current coordinates (such as 3,4,5) into the equation and get:

enter image description here

So, this new vector (1, 8, 75) would be the direction we’d move in to increase the value of our function.

The other use of gradient is to find the max/min of multivariable functions.

vidhan
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For a multivariable function $f(x,y)$, that vector is called $\textbf{grad}f$, the gradient of $f$, sometimes denoted $\nabla f$.

It's a vector that points in the direction of greatest increase of the function.

jonbaldie
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    The statement: "think of ∇f as a normal vector pointing perpendicularly out of the surface." is incorrect. When considering f(x,y), you should rather think of ∇f as a vector field projected into the plane, as depicted here: http://en.wikipedia.org/wiki/Gradient in the third figure from above. – Cyclone May 18 '15 at 09:34