Vector Calculus Independent Study

Unit 1: Vectors and Three Dimensional Geometry

In this unit you should/will learn:

1. Points and Vectors in 3 space. • How to associate cartesian coordinates to a point in R^{3}.

• How to find the distance between two points (Pythagorean theorem).

• The definition of a vector, and the meaning of 0, i⃗, j⃗, and k⃗.

• How to find the length of a vector, |v⃗|.

• How to find the vector between two points, v⃗_{a,b} = b⃗ − a⃗.

• How to add and subtract vectors. Geometric meaning.

• How to multiply/divide a vector by a constant. Geometric mean-

ing.

2. Lines in 3 space.

• Parametric definition: l(t) = p⃗ + tv⃗

• How to find the line between 2 points.

• How to find the vector lying along the line (v⃗).

• How to find the line going through a point along a given vector.

• How to convert between parametric and implicit definitions of a line. (Implicit is A_{1}x+B_{1}y+C_{1}z+D_{1} = 0, A_{2}x+B_{2}y+C_{2}z+D_{2} = 0).

3. Planes in 3 space.

• Implicit definition: Ax + By + Cx + D = 0.

• How to find the plane containing 3 points.

• How to find the plane containing a line and a point.

• How to find the plane containing 2 intersecting lines.

• How to find the normal to a plane. (A, B, C).

• How to find the plane through a point with a given normal vector: A(x − x_{0}) + B(y − y_{0}) + C(z − z_{0}) = 0.

• How to convert between implicit and parametric P (r, s) = p⃗ + rv⃗_{1} + sv⃗_{2}. (p⃗ is a point on the plane, v⃗_{1} and v⃗_{2} are vectors lying on the plane.)

4. Dot Products.

• Algebraic definition: a⃗ · b⃗ = a_{1}b_{2} + a_{2}b_{2} + a_{3}b_{3}.

• Geometric definition: a⃗ · b⃗ = |a||b| cos θ.

• How to find the angle between 2 vectors using the dot product.

• How to project a vector onto another vector.

• How to project a point onto a line.

• How to find the distance between a point and a line.

• How to test if two vectors are perpindicular (a⃗ · b⃗ = 0 ?)

5. Cross Products.

• Geometric definition: |a⃗ × b⃗| = |a||b| sin θ, with direction given by right hand rule.

• Algebraic definition: a⃗×b⃗ = det [ \begin{matrix}i⃗ & j⃗ & k⃗ \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} ] = (a_{2}b_{3} −b_{2}c_{3})i⃗−

(a_{1}b_{3} − a_{3}b_{1})j⃗ + (a_{1}b_{2} − a_{2}b_{1})k⃗

• How to find a vector perpendicular to two other vectors.

• How to find the area of a parallelogram spanned by two vectors.

• How to find the area of a triangle spanned by two vectors.

• How to find the plane going through two given vectors.

6. Other Coordinate Systems • The definition of the cylindrical coordinate system (r, θ, z).

• The definition of the spherical coordinate system (ρ, θ, ϕ).

• How to convert a point in cartesian coordinates to cylindrical or spherical coordinates, or vice-versa.

• How to convert an equation in cartesian coordinates to cylindrical or spherical coordinates, or vice-versa.

Suggested Procedure:

1. Read and do some problems from • Rogers Chapters 1, 2, 3, 4, and sections 7.1, 7.2, and 17.1.

• Marsden and Tromba, chapter 1. [Section 1.5 is optional.]

• Thomas & Finney chapter 13, or

• Simmons section 17.3, and chapter 18.

2. Take the sample test.

3. Take a unit test.