Multivariable Functions and Surfaces

Recall the 3D Cartesian Coordinate System:

3D Coordinate System

• Description of 3D space – analogous to 2D
• 3D space: ${\mathbb{R}}^3$, 2D space: ${\mathbb{R}}^2$

• Distance between 2 points: $d(P_1,P_2)=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z1{)}^2}$
• Equation for a sphere: $(x-h{)}^2+(y-k{)}^2+(z-l{)}^2=r^2$

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We use right-handed coordinate systems:

Try to keep these axes in this orientation, or answers may come out negative.

Definition: Octants

We split 3D Cartesian space into 8 regions called octants.

• Analogous to quadrants for 2D Cartesian space.

Octants are collections of points $(x,y,z)$ where $x,y,z$ have some given signs.

• First octant is the region where $x,y,z$ are positive.