Calculus Volume 3 · 免费预览
偏导数
偏导数是在将除一个变量之外的所有变量保持不变的情况下，沿单一方向求导；在连续性条件下，高阶偏导数可以交换次序，混合偏导数相等是本节著名的结果。
1. Definition and computation
Definition and computation
2. Higher partials and Clairaut
Higher partials and Clairaut
3. Interpretations and applications
Interpretations and applications
4. The partials' information content
The partials' information content
5. Partial derivative computation
For f(x, y) = x^2 y + sin(xy), the partial f_x equals which expression?
6. Geometric meaning
Geometrically, f_x(a, b) measures which slope?
7. Clairaut theorem
Clairaut's theorem states which equality, under continuity of the partials?
8. Holding constant
To compute f_x of f(x, y) = x^3 y^2, which operation does one perform?
9. Partial versus total
A hiker walks a path x = x(t), y = y(t) on the hillside f(x, y). The rate df/dt is the total derivative, combining f_x and f_y with which other factors?
10. Mixed partial count
A function of two variables has two first partials and three distinct second partials under Clairaut. As a number, how many distinct second partials? Answer with a number.
11. Partial notation symbol
The partial derivative symbol is a curly d. As a single character, what is it? Answer with the character.
12. Heat equation order in time
The heat equation has the first time derivative on the left. As a number, what order? Answer with a number.
13. Cobb-Douglas marginal
For the Cobb-Douglas production P = c L^alpha K^(1-alpha), the marginal product of labor dP/dL equals which expression?
14. Wave equation second
The wave equation is second order in both variables. As a number, how many second derivatives appear? Answer with a number.