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Illustrative quantitative study room with chalkboard curves and mathematical notation, an open book and brass instruments in warm light.
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Give an idea a shape.

Move from intuition to a model. Examine its assumptions, then work through the consequences.

FOLLOW THE MECHANISMA surface that explains its own motion
STARTING CHOICE
J=0.35(x2+y2)J=0.35(x^2+y^2)
(1.5, 1.0)

Two inputs become one surface height.

DIRECTION
v=−∇J\mathbf v=-\nabla J
(-1.050, -0.700)

Negative gradient: move against the slope.

CURRENT OUTCOME
Jt=J0e−1.4tJ_t=J_0e^{-1.4t}
J = 1.137

This is the height of the moving point.

01 / A PAIR BECOMES A HEIGHT

The selected point is your pair of inputs. Its height is J = 0.35(x² + y²).

Starting penalty J₀1.14
Starting slope in x+1.05
THE CONNECTION

The surface point, the differential equation and the decay curve are three views of the same model. Change the starting coordinates to see all three respond.

Go deeper: read the mechanism and definitions

The negative gradient is (−0.70x, −0.70y). Solving these differential equations gives exponential decay in both coordinates. Squaring those coordinates makes the penalty decay twice as fast. The surface, selected point and chart use these exact functions; there are no simulated price claims.

x(t)=x0e−0.7t,y(t)=y0e−0.7tx(t)=x_0e^{-0.7t},\quad y(t)=y_0e^{-0.7t}

This is the exact function drawn in the surface above. Its Hessian is 0.70 times the identity matrix, so the surface curves upward in every direction and has a unique minimum at (0, 0). A sufficiently small step opposite the gradient lowers the penalty. Doubling both deviations from the origin multiplies J by four.

Normalized teaching example. This is a mathematical penalty surface, not an estimated portfolio-risk model.

Interactive quadratic surface · hypothetical inputs

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2 courses

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