Deeper

Lab · c10.s04.L1

Gradient field

At a point on the bowl the gradient arrow points uphill, crosses the level curve at a right angle, and its length is the steepest slope there.

Question: Which way is uphill?

Gradient fieldA contour map of the tilted bowl f(x, y) = 0.5x^2 + y^2 + 0.5xy, with five labelled gold level curves and a pale grid of small arrows. A russet arrow grows from a probe point. At every step the arrow crosses the gold curve through the point at a right angle, points uphill, and its length equals the steepest slope; a strip below plots slope against direction, peaking at the arrow's angle.Gradient fieldf(x, y) = 0.5*x^2 + y^2 + 0.5*x*ygrad f = (x + 0.5*y, 0.5*x + 2*y)0.51.01.52.54.0xySlope in each directionsame point, one full turn-8-40480°90°180°270°360°direction (degrees)1.803Probe (2.0, -1.0) · height f = 2.000 · level 2.000Arrow 1.803 = steepest slope at -33.7° · along the contour 0.000
Re-simulated Re-simulated: the surface f(x, y) = 0.5*x^2 + y^2 + 0.5*x*y and its gradient grad f = (x + 0.5*y, 0.5*x + 2*y) are computed exactly by data/scripts/c10.s04.L1.py, so nothing here is measured. The nine frames step a probe through stored lattice points; at each one the russet uphill arrow, the gold level curve through the probe and the slope strip are read from that point's record.
Lab controls
2

Left or Down one step, Right or Up one step, Page Down or Page Up two steps, Home -2.5, End 2.5

-1

Left or Down one step, Right or Up one step, Page Down or Page Up two steps, Home -2.5, End 2.5

Frame 1 of 9: Where the probe startsThe probe sits at (2.0, -1.0), where the height is 2.000. The russet arrow points uphill at -33.7° and is 1.803 long, the steepest slope there. The gold curve through the point is the level f = 2.000, and the arrow meets it at a right angle.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

A contour map of the tilted bowl f(x, y) = 0.5x^2 + y^2 + 0.5xy, with five labelled gold level curves and a pale grid of small arrows. A russet arrow grows from a probe point. At every step the arrow crosses the gold curve through the point at a right angle, points uphill, and its length equals the steepest slope; a strip below plots slope against direction, peaking at the arrow's angle.

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Caption status: re-simulated. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.