What does a function of two variables assign to each ordered pair (x,y)?
A unique real number f(x,y).
What is an n-tuple in the context of functions of several variables?
An ordered set of n elements used as input.
What is the domain of a multivariable function?
The subset of ℝ² where the function is defined.
Why is the domain of all pairs with ?
Because the denominator cannot be zero.
Function of Two Variables - Definition
Assigns a real number to each (x,y) in its domain.
Domain of Function
Subset of ℝ² where function is defined.
Limit in Multivariable
Approaches L along every path to (x₀,y₀).
Continuity
Function value equals limit at (x₀,y₀).
Partial Derivative
Derivative treating one variable as constant.
Total Differential
dz = f'_x dx + f'_y dy.
Second Partial Derivatives
Partial derivatives of partial derivatives.
Answer 10 questions on Multivariable Functions and Optimization.
1. What does a function of two variables assign to each ordered pair in its domain?
2. Which expression represents an ordered set of n input elements for a function of several variables?
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