Multivariable Calculus · 免费预览

多变量函数

实值二元函数和三元函数、它们的定义域和值域，以及描绘它们的两种标准方式：把图像看作曲面，把等值集看作曲线。

1. A function of several variables assigns one number to each point

A function of several variables assigns one number to each point

2. The graph lives one dimension higher than the input space

The graph lives one dimension higher than the input space

3. Level sets slice the graph at constant heights

Level sets slice the graph at constant heights

4. Distinguishing the graph from the level sets

Distinguishing the graph from the level sets

5. Reading the natural domain

What is the natural domain of the function f(x,y)=1x−yf(x, y) = \dfrac{1}{x - y}?

6. A level set of the bowl

For f(x,y)=x2+y2f(x, y) = x^2 + y^2, what is the level set at value c=4c = 4?

7. Level sets of a saddle

The level sets of f(x,y)=xyf(x, y) = xy are which family of curves?

8. The boundary of a square-root domain

Give the radius of the largest circle centered at the origin that fits entirely inside the domain of g(x,y)=9−x2−y2g(x, y) = \sqrt{9 - x^2 - y^2}, as a plain number.

9. The range of a square-root function

For g(x,y)=9−x2−y2g(x, y) = \sqrt{9 - x^2 - y^2}, give the largest value in the range, as a plain number.

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