Question

Sketch (as best you can) the graph of the monkey saddle \(z=x\left(x^{2}-3 y^{2}\right) .\) Begin by noting where \(z=0\)

Short answer

The monkey saddle intersects the xy-plane along the y-axis, \(x=\pm\sqrt{3}y\).

Step-by-step solution

Understanding the Equation

The given equation is \( z = x(x^2 - 3y^2) \). This is a surface in the three-dimensional space. To sketch this, let's understand what the equation represents. It's a polynomial in terms of \( x \) and \( y \) that defines the height \( z \).

Finding Where z = 0

To find where the surface intersects the \( xy \)-plane, set \( z = 0 \). The equation becomes \( x(x^2 - 3y^2) = 0 \). This gives two factors: \( x = 0 \) and \( x^2 = 3y^2 \). The first, \( x = 0 \), represents the whole \( y \)-axis, while the second, \( x^2 = 3y^2 \), represents the lines \( x = \sqrt{3}y \) and \( x = -\sqrt{3}y \).

Analyzing the Surface

The surface is called a 'monkey saddle' because it has more than the usual saddle points. At \( x = 0 \), notice that it intersects along the \( y \)-axis. On lines \( x = \sqrt{3}y \) and \( x = -\sqrt{3}y \), the surface again intersects the \( xy \)-plane.

Visualizing the Curvature Along Axes

Along the \( x \)-axis (\( y = 0 \)), the equation simplifies to \( z = x^3 \), which is a cubic function intersecting the plane at the origin. Along the \( y \)-axis (\( x = 0 \)), \( z = 0 \) which is a flat line at the origin, confirming the intersection found earlier. Evaluate curvature using derivatives or recognize the symmetry.

Sketching the Graph

To sketch the graph, draw the \( xy \)-plane, then plot the \( y \)-axis, \( x = \pm \sqrt{3}y \) lines. Consider the curvature along these paths and notice that surface dips to negative or rises to positive at different sections based on \(x \) and \( y \) relationships. The surface has a 'saddle point' effect at the origin.

Key concepts

Three-dimensional surfaces

Three-dimensional surfaces are fascinating geometric shapes that exist in the realm beyond our regular two-dimensional sketches. In mathematics, these surfaces are defined by equations that involve three variables, typically denoted as \(x\), \(y\), and \(z\). The variable \(z\) often represents the elevation or height above the \(xy\)-plane. An excellent example is the equation of a monkey saddle: \(z = x(x^2 - 3y^2)\).

When dealing with three-dimensional surfaces, the idea is to understand how the heights change across a plane. Visualization techniques such as contour plots, wireframes, or even software tools can help in visualizing these surfaces.

Surfaces can have complex shapes and features, such as peaks, valleys, and saddle points, all influencing its geometry. By setting the surface equation to equal zero, we can find intersections with the \(xy\)-plane, which are useful for sketching and understanding the surface's orientation and behavior.

Polynomial equations

Polynomial equations form the backbone of surface definitions in three dimensions. These equations consist of sums of terms, each made by multiplying a constant and a variable raised to a power. The monkey saddle's equation is a polynomial because it includes multiplies of variables \(x\) and \(y\) raised to various powers.

Understanding polynomial equations involves recognizing these power structures and their effects on the overall shape. For instance, in \(z = x(x^2 - 3y^2)\), the \(x^2\) term is quadratic, showing symmetry along the \(x\)-axis and creating a characteristic saddle shape. Meanwhile, \(3y^2\) alters this shape by introducing a second dependency along the \(y\)-axis.

Solving polynomials involves finding values of variables that satisfy the equation, like setting \(z = 0\) in the monkey saddle. This practice helps discover lines or points where the surface intersects or makes specific geometric features evident.

Saddle points

Saddle points are places on surfaces where the local shape resembles a saddle. These points are neither complete maxima nor minima. They are interesting features because they represent where the surface curves upward in one direction and downward in another.

The monkey saddle has a prominent saddle point at the origin \((0,0)\). This feature is because at \(x = 0\) and \(y = 0\), the polynomial's terms align uniquely to form a flat point with changes in curvature in different directions.

To analyze saddle points, you can take partial derivatives of the polynomial function. These derivatives help in assessing how the function varies along the \(x\) and \(y\) axes. For the monkey saddle, examining such changes confirms the surface's complex nature, characterized by twisting and turning at its saddle point.