$\bigskip $Directional derivative



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The graph of $z=x^{2}+y^{2}$ with the tangent at MATH in the direction of $3i+4j$

The slope of the red line is the value of the derivative of $z=x^{2}+y^{2}$ in the direction of the vector $3\vec{i}+4\vec{j}$ at the point MATH which isMATH

$\bigskip $For a unit vector MATH we can calculate MATH byMATH




The Gradient

The gradient of MATH is the vectorMATHIt has the following important properties:

$\bigskip $

$\bigskip $Below are graphs, contour diagrams and gradients of several functions.

Note that:




For linear functions, all the contours are parallel and equally spaced. Also, the gradient vector is the same at every point.

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$z=2x-y+1$
graphics/2D__23.png graphics/2D__24.png




The graph of $z=x^{2}+y^{2}$ , its contour diagram and several of its gradient vectors. Notice that the distances between the circles are shrinking and the gradient vectors are growing in size as you get further away from the origin.




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The graph of $z=x^{2}+y^{2}$
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Countours with MATH
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The graph of MATH , its contour diagram and several of its gradient vectors.




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The graph of MATH
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Countours with MATH
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MATH

The graph of MATH ,its contour diagram and its gradient vectors. Note the distances between the circles growing are and the gradient vectors are shrinking in size as you get further away from the origin.

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The graph of MATH
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Countours with MATH
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MATH




The graph of $z=xy$ ,its contour diagram and its gradient vectors.

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graph of $z=xy$
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Contour diagram for $z=xy$
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MATH

$.$





The gradient MATH gives a vector for every point MATH.

It is an example of a vector field. (Math 324 or Math 334)

Can you tell what a function looks like by looking at its gradient vectors?




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MATH
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$z=5x-7y+4$




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MATH
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$z=x^{2}$







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MATH
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MATH


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The vector at a given point $\left( x,y\right) $ is $y\vec{i}-x\vec{j}$ $.$ It is not the gradient of any function.