3-D surfaces

Section: Understanding 3-D Surfaces

• 3-D surfaces represent three-dimensional geometric spaces and are often visualised in terms of x, y, and z coordinates.

• A surface is defined by an equation of the form z = f(x, y). This is the standard form for a 3-D surface.

• Sometimes a surface might be expressed in the form F(x, y, z) = 0. This is called an implicit equation.

• Familiarise yourself with common types of surfaces such as planes, cylinders, cones, and spheres.

Section: Graphing 3-D Surfaces

• Graphing these surfaces requires skill in 3-D spatial awareness.

• You can often visualise the surface as a contour map, which shows constant Z-values in the X-Y plane.

• Positive and negative regions on the surface are often distinguished by shaded and unshaded areas, respectively.

• Be aware of key features like intercepts (where the graph meets the x, y, or z axis) and symmetries as they can help to sketch the graph correctly.

• Surfaces can also be recognised and sketched by comparing them with known surfaces such as paraboloids, hyperboloids, ellipsoids, etc.

Section: Using Derivatives in 3-D Surfaces

• Partial derivatives evaluate the gradient of a surface along the x or y direction while keeping the other variable constant. They are denoted as ∂f/∂x or ∂f/∂y.

• Second order partial derivatives examine the change in the respective first order derivative.

• A surface reaches a local maximum or minimum at a point where the first order derivatives are zero and the second order derivatives have the same sign.

• Partial derivatives can also be used to discover a surface’s tangent plane at a certain point.

• The equation of this plane can be found using the first order derivatives at the specified point.

• A gradient vector can be defined, which gives the direction of the steepest ascent on the surface from a given point.

• Techniques involving the change of variables can be applied to equations of 3-D surfaces.