Calculus Volume 3 · 免费预览
方向导数与梯度
方向导数衡量任意方向上的变化率，即梯度与方向单位向量的点积；梯度指向上升最陡的方向，其大小为最陡斜率，并与等值线垂直。
1. Directional derivatives
Directional derivatives
2. The gradient
The gradient
3. Properties and applications
Properties and applications
4. The gradient in three variables
The gradient in three variables
5. Directional derivative definition
The directional derivative D_u f at (a, b) measures which rate?
6. Gradient components
The gradient of f(x, y) is which vector?
7. Steepest direction
The direction of greatest rate of increase of f at a point is which?
8. Level curve perpendicularity
The gradient at a point is perpendicular to what?
9. Unit direction requirement
The directional derivative formula D_u f = gradient dot u requires u to be what?
10. Gradient symbol name
The gradient operator's symbol is an inverted delta. As a single word, what is it called? Answer in lowercase.
11. Steepest descent direction
Minimizing steps opposite the gradient. As a sign, which? Answer with the single character.
12. Directional as dot product
D_u f equals the gradient dot which vector? Answer with the single letter u.
13. Zero directional rate
The directional derivative is zero in which direction, when the gradient is nonzero?
14. Gradient descent use
Gradient descent, stepping opposite the gradient, is a foundational algorithm of which field? Answer with two words.