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MA35 Labs

1 Single Variable Calculus

2 » Two Variable Calculus

3 Three Variable Calculus

Polar Coordinate Labs

Two Variable Functions

Contents

2.1 Functions of Two Variables

2.2 Differentiation

2.3 More Derivatives

2.3.1 Directional Derivative

2.3.2 Second Partial Derivatives

2.3.3 Hessian Determinant

2.3.4 Taylor Series

2.3.5 Constrained Maxima & Minima

2.3.6 Equality of Mixed Partials

2.4 Integration

2.5 Vector Analysis

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Hessian Determinant

Text

The Hessian determinant of a function f(x,y) is defined as H(x,y) = f_xx(x,y)f_yy(x,y) - f_xy(x,y)f_yx(x,y).

The Second Partials Test states that if a function f(x,y) has continuous second partials and f_x(x₀,y₀) = 0 and f_y(x₀,y₀) = 0, then

1. H > 0 and f_xx(x₀,y₀) > 0 implies (x₀,y₀) is a local minimum;

2. H > 0 and f_xx(x₀,y₀) < 0 implies (x₀,y₀) is a local maximum;

3. H < 0 implies (x₀,y₀) is a saddle point;

4. H = 0 then the test is inconclusive.

Demos

Hessian determinant

In this demo, we color the surface according to the value of the Hessian at every point. The Hessian exists at every point and serves two purposes. One purpose is to tell us about the shape of a surface. A region of a surface where the Hessian is positive will have a shape resembling that an elliptic paraboloid (bowl-shaped) and a region with a negative hessian will have a shape resembling that of a hyperbolic paraboloid (saddle-shaped). A Hessian of 0 means the shape is neither like that of an elliptic paraboloid nor like that of a hyperbolic paraboloid. The other purpose of a Hessian is to tell us what kind of a critical point we have if the first partial derivatives are equal to 0. If a given critical point is located in a neighborhood colored in orange, then the Hessian at this critical point is positive, and it has to be a maximum or minimum. If the critical point is in a green neighborhood, then the Hessian is negative, so the critical point is a saddle. If the critical point is in a white neighborhood, then the Hessian is zero and doesn't give any conclusive information.

Hessian determinant
	This example shows how the sign of the Hessian affects the shape of a region of a surface.

Exercises