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Gradient Fields:

Our gradient f , f , ( ) , , x y f x y f f =

“dell f ” from 11.6 are vector fields. partial derivatives

( ) , , , , x y z f x y z f f f = =

Gives the gradient vector at each point (x, y) .

Remember Figure 15 at the beginning of the lesson showing f is perpendicular to the two- dimensional level curves? Gradient vectors are perpendicular to the level curves. f is also

perpendicular to the three-dimensional level surfaces (perpendicular to the tangent plane at the point of tangency). If they’d like to see more visuals, show them p.795 Fig.7, p.796 Fig.9, p.797 Fig. 10, and p.798 Fig. 11.

Reminder: gradient vectors are long where the level curves are close together (short where the level curves are farther apart). Closely spaced level curves indicate a steep graph (steep graph means a greater slope (value), so a larger vector). The length of a gradient vector is the value of the directional derivative of f.

I n o r d e r t o s k e t c h t h e g r a d i e n t v e c t o r f i e l d f o f f , f i r s t f i n d ( ) , , x y f x y f f = a n d t h e n s k e t c h i t t same way we did HW #4. h e

Let’s use Mathematica on this next one so you can do #27 on HW. Example 1:

Find the gradient vector field f of f (x, y) = 1 (x + y)2 . 4

We are going to plot the gradient vector field along with the contour plots.

[ ] ( ) 1 4 , : , 2 f x y x y


o n t o u r

P l o t [ f

] { , , , x y x

} ( 3 , 3 , , y

) 3 , 3

c o n t o u r , s h a d i n g F a l s e

If you’re standing on a contour line and walk perpendicular () to it, you’ll increase () or decrease () the greatest.

Needs [“Vector Field Plots”] ⟨⟨ Vector Field Plots

G r a d i


t F i e l d P l o t [ f

] { , , , x y x

} { 3 , 3 , , y

} 3 , 3

Show [%, % ,%]

You’ll do HW #27 using Mathematica. Just bring up my example and adjust.

***Notice the gradient vectors are orthogonal to the contour lines!

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