# MA 227 Standard C01

Surfaces in Three-Dimensional Space

At the end of the course, each student should be able to…

• C01: SurfaceEQ. Identify and sketch surfaces in three-dimensional Euclidean space.

## C01. Surfaces in Three-Dimensional Space

• Simple planes parallel to coordinate axes are given by $$x=a$$, $$y=b$$, and $$z=c$$.
• The dot product may be used to obtain equations for arbitrary planes.
• The equation $$\vect N\cdot(P-P_0)=0$$ describes the points $$P$$ on the plane passing through the point $$P_0$$ and normal to the vector $$\vect N$$.
• If $$P=\<x,y,z\>$$, $$P_0=\<x_0,y_0,z_0\>$$, and $$\vect N=\<A,B,C\>$$, then this equation may be expressed as $$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$$ and simplifies to $$Ax+By+Cz=D$$ for some value of $$D$$.
• Quadric surfaces are another common example used in calculus.
• These surfaces are obtained from quadratic equations (polynomials of degree $$2$$) of the variables $$x,y,z$$.
• The simplest example is that of the sphere $$\|P-P_0\|=r$$ describing the points $$P$$ that are distance $$r$$ from the sphere’s center $$P_0$$.
• If $$P=\<x,y,z\>$$, $$P_0=\<x_0,y_0,z_0\>$$, and $$\vect N=\<A,B,C\>$$, then this equation may be expressed as $$(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2$$.
• The surfaces given by equations of $$x,y,z$$ may be sketched by considering traces of the surface in simple planes $$x=a$$, $$y=b$$, or $$z=c$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 11.1 (exercises 1-30, 41)
• 11.5 (exercises 21-26)
• 11.6 (exercises 1-12, 33-44)