Vector-Valued Functions of Several Variables

Question 1

How do you solve an Epsilon-Delta Equation for limit?

Answer 1

Assume a value for Delta which the independent variables (usually x or x0) differ from, and show that this directly proves that the dependant variable and the limit also differ from some number Epsilon, i.e.
0 < ∥x − x0∥ < δ ⇒ ∥f(x) − l∥ < ϵ

Question 2

What does it mean for function f to be a linear mapping?

Answer 2

f(x + h) = f(x) + f(h).

Question 3

What is the name of the type of matrix used to solve a linear mapping derivation?

Answer 3

A Jacobian Matrix.

Question 4

What does element a(i,j) of a Jacobian Matrix look like?

Answer 4

∂fi/∂xj

Question 5

When would one use ∂ instead of d in differentiation?

Answer 5

When differentiating a function of more than 1 variable, or to signify partial differentiation.

Question 6

What would you write to represent a Jacobian Matrix?

Answer 6

A or Df(x)

Question 7

The i’th row of A is the ____ of fi.

Answer 7

The gradient of fi (∇fi).

Question 8

What must one do to use the chain rule on functions of more than one variable?

Answer 8

When differentiating, find the partial derivative of the function with more than 1 variable. This is also true if all functions have more than one variable

Question 9

What does altering the meaning of a differential operator mean?

Answer 9

When working through equations involving the chain rule, you may occasionally have to put values in terms of other variables. Finding out the value of a differential operator, i.e.
∂/∂x, in terms of these initial variables can be useful (like substitution in integration!)