Visualize partial derivatives, gradient vectors, and double integrals on 3D surfaces with interactive cross-sections.
A typed two-variable function becomes a surface z = f(x,y), with movable slices that isolate change in the x or y direction. At a selected point, the visualizer relates those cross-section slopes to the partial derivatives and gradient, and it can show a double integral over chosen bounds. The common surprise is that the gradient belongs to the x-y input plane: it points toward the fastest increase of f, while the surface is drawn in three dimensions. A double integral is signed, so regions below z = 0 subtract rather than add geometric volume.
A concrete input and expected output from the current implementation.
Input
f(x,y) = x^2 + 2y; point = (1,3)
Expected output
Surface height: 7; df/dx = 2; df/dy = 2; gradient = (2, 2).
At (1,3), the function value is 1^2 + 2(3) = 7. Since df/dx = 2x and df/dy = 2, their values at that point are 2 and 2, which form the gradient (2, 2).
Fixing y and varying x traces an x-direction curve through the surface; its local slope is df/dx. Fixing x and varying y gives df/dy. The gradient combines those values as (df/dx, df/dy) at the selected input. Its direction is steepest ascent under the ordinary Euclidean coordinate scale, and its magnitude is the maximum directional derivative there.
Lower and upper bounds for x and y identify the region used by the double-integral view. Surface heights over that region are accumulated with their signs: positive heights contribute positively and negative heights contribute negatively. For a continuous function on a rectangular region, changing the order of integration does not change the final value, although the nested setup and cross-sections differ.
No. A zero gradient identifies a critical point when the relevant partial derivatives exist, but it does not classify that point. The surface may have a local minimum, local maximum, saddle, or another inconclusive shape there. Inspect the surrounding graph and use additional analysis, such as second derivatives or directional behavior, before assigning a classification.
The displayed integral is signed accumulation, not necessarily unsigned volume. Enlarging the bounds may include surface values below z = 0, whose negative contributions reduce the total. Cancellation can therefore make a larger region yield a smaller value, zero, or a negative value. If actual geometric volume is needed, verify that the function stays nonnegative or integrate its absolute value where appropriate.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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