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Partial Derivatives and Gradients

A surface has a slope in every direction. Slice it along x and along y to get the two partial derivatives; together they make the gradient, the arrow of steepest climb.

Step 1 of 21

The surface z = sin x + cos y

Contour map, seen from above

— x-slice (y fixed): slope ∂f/∂x— y-slice (x fixed): slope ∂f/∂y→ ∇f, steepest climbcontours: low to high

Speed

z = sin x + cos y: a landscape of hills and hollows. Its contour map, on the right, shows the same thing from above — each ring is a height. P sits at (1, −0.6), height 1.667.

f(x, y) = \sin x + \cos y, \qquad f(1, -0.6) = 1.667

Partial Derivatives and Gradients

A surface has a slope in every direction. Slice it along x and along y to get the two partial derivatives; together they make the gradient, the arrow of steepest climb.

Step 1 of 21

The surface z = sin x + cos y

Contour map, seen from above

— x-slice (y fixed): slope ∂f/∂x— y-slice (x fixed): slope ∂f/∂y→ ∇f, steepest climbcontours: low to high

Speed

z = sin x + cos y: a landscape of hills and hollows. Its contour map, on the right, shows the same thing from above — each ring is a height. P sits at (1, −0.6), height 1.667.

f(x, y) = \sin x + \cos y, \qquad f(1, -0.6) = 1.667

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Partial Derivatives and Gradients

See also

Partial Derivatives and Gradients

See also