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The below graph illustrates the relationship between the level curves and the graph of the function. The key point is that a level curve $f(x,y)=c$ can be thought of as a horizontal slice of the graph at height $z=c$.
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Graph of elliptic paraboloid by Duane Q. Nykamp is licensed...
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Level curves of the function g(x,y)=√9−x2−y2 g (x y) = 9 − x 2 − y 2, using c=0,1,2 c = 0 1, 2, and 3 3 (c=3 c = 3 corresponds to the origin). A graph of the various level curves of a function is called a contour map.
Imagine taking this plot of level curves, twisting clockwise by 135 degrees (by symmetry this will be the same as rotating 45 degrees counter-clockwise) and then pulling the bottom (left) corner towards yourself. The green curve at k=0 corresponds exactly to the trough in the graph. Now check-out the cross-section at x=2 . In Maple type.
Recall from Section 15.1 that the curve. f(x,y)=. z. 0. , where. z. 0. is a constant, is a level curve, on which function values are constant. Combining these two observations, we conclude that the gradient.
Graphs and Level Curves. Read Lesson 10 in the study guide. Read Section 12.2 in the text. Continue work on online homework Also Try 11, 15, 21, 25, 27, 31, 33, 43, 47. Mth 254H – Winter 2013. Examples. (x, y ) = cos(y )esin x. (x, y ) = 4. 1/7. x2 y 2. 2. 0. -2 -4 -2. y 2. x. 2. 4 4. Mth 254H – Winter 2013. 4. 3. 2. 1. 0 -2. -1 t. 1. r. 1 22. 3/7.
There is a close relationship between level curves (also called contour curves or isolines) and the gradient vectors of a curve. Indeed, the two are everywhere perpendicular.
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