Lines and Planes in Space

Equations of Lines

In three-dimensional space, a line can be described using vectors and parametric equations. These forms are essential for understanding the geometry of lines and their relationships.

• Vector Equation: The line through point with direction vector is given by:

• Parametric Equations: where is a real parameter.

• Symmetric Equation:

Equations of Planes

A plane in space can be defined by a point and a normal vector.

• General Equation: The plane through with normal vector :

Relative Position of Lines

• Two lines in may be parallel, intersecting, skew, or coincident.

• Skew lines do not intersect and are not parallel.

• If two lines are in the same plane, they are either parallel or intersecting.

Distance Calculations

• Distance from a Point to a Plane: The shortest distance from point to plane is:

Vectors: Position, Velocity, and Acceleration

Vector-Valued Functions

Describes motion in space using functions of time.

• Position Vector:

• Velocity Vector:

• Acceleration Vector:

• Chain Rule for Vector Functions: If , then

Functions of Several Variables

Level Sets and Graphs

Functions or can be visualized using level curves or surfaces.

• Level Curve: Set of points where for constant .

• Level Surface: Set of points where .

Limits and Continuity

• Limit Definition: if for every , there exists such that whenever .

• Continuity: is continuous at if .

• To show a limit does not exist: Find two paths to the point that yield different limits.

Partial Derivatives and Differentiability

• Partial Derivative:

• Differentiability: is differentiable at if partial derivatives exist and can be well approximated by a linear function near .

• Gradient Vector:

• Directional Derivative: In direction :

Tangent Lines and Critical Points

• Tangent Line to Level Curve: At , the tangent line to is perpendicular to .

• Critical Point: Where .

• Second Derivative Test: For at , compute :

  • If and , local minimum.

  • If and , local maximum.

  • If , saddle point.

  • If , test is inconclusive.

Sequences and Series

Sequences

A sequence is a list of numbers in a specific order, often defined by a formula.

• Definition: is a sequence if is defined for each .

• Limit of a Sequence: if for every , there exists such that for all .

• Sandwich Theorem: If and , then .

• Order of Growth: Exponential Polynomial Logarithmic

Series

A series is the sum of the terms of a sequence. Convergence depends on the behavior of the sequence of partial sums.

• Partial Sum:

• Convergence: converges if exists and is finite.

• Tests for Convergence:

  • Integral Test: If is positive, decreasing, and converges, then converges.

  • Ratio Test: If , converges absolutely; , diverges; , inconclusive.

  • Root Test: Same conclusions as ratio test.

  • Alternating Series Test: If decreases to $0\sum (-1)^n a_n$ converges.

• Absolute vs. Conditional Convergence: converges absolutely if converges.

Power Series

• General Form:

• Radius of Convergence: The series converges for .

Taylor and Maclaurin Series

Taylor Series

The Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point.

• General Formula:

• Maclaurin Series: Taylor series at .

• Common Maclaurin Series:

  • ,

• Remainder Estimate: If for in interval , then the error in approximating by its th Taylor polynomial is at most:

Vectors and Geometry of Space

Distance Between Points

• Distance between and :

Dot Product

• Definition:

• Angle Between Vectors:

Cross Product

• Definition:

• Magnitude:

• Geometric Meaning: Area of parallelogram spanned by and .

• Properties:

  • Anticommutative:

  • Distributive:

• Volume of Parallelepiped:

Summary Table: Series Convergence Tests

Example Applications

• Finding the equation of a plane: Given point and normal vector , the plane is .

• Radius of convergence for power series: For , the radius is infinite.

• Critical points: For , , so the only critical point is , which is a minimum.

Additional info: These notes cover topics from Calculus II and III, including multivariable calculus, sequences and series, and vector calculus, matching chapters 1, 2, 3, 4, 5, 10, and 11 from the provided list.