Vector Calculus

Question 1

What does ℝ stand for?

Answer 1

The set of real numbers

Question 2

What does ℝⁿ stand for?

Answer 2

Set of ordered n-tuples (x₁, x₂, … , x_n) where each x_i is real, and can be thought of as cartesian coordinate of a point in n-dimensional space

Question 3

What is the standard basis?

Answer 3

{e₁, e₂, …, e_n}

Question 4

The position vector of a point in ℝⁿ can be written in the standard basis as what?

Answer 4

Question 5

What drops the sum symbol when we see a repeated index?

Answer 5

The Einstein Summation Convention

Question 6

What are the e_i’s w.r.t the scalar (dot) product?

Answer 6

Orthonormal

Question 7

What does the following equal.

Answer 7

Question 8

What is δ_ij called?

Answer 8

The Kronecker delta.

Question 9

What is the scalar (dot) product of two vectors u and v?

Answer 9

Question 10

What is the equation for the length of a vector u?

Answer 10

Question 11

What is the equation for the angle θ between u and v?

Answer 11

Question 12

Define a scalar field.

Answer 12

Question 13

Define a vector field.

Answer 13

Question 14

How are curves in ℝⁿ given?

Answer 14

Question 15

If x(t) is differentiable then what is the tangent to the curve?

Answer 15