Gradient: Geometry, Coordinates, and Differentiation

All flashcards

What is the geometric meaning of the gradient?

For a differentiable scalar function, it is a vector pointing in the direction of fastest increase.

What does $‖∇f‖$ measure?

It is the greatest directional rate of increase per unit distance at the point.

What is a directional derivative?

It measures the instantaneous rate at which a function changes when moving in a specified direction; for a unit vector $v̂$, it equals $∇f·v̂$.

What is a stationary point?

A point where the gradient is the zero vector. It may be a maximum, minimum, saddle point, or another type of stationary point.

How does the gradient describe a terrain surface?

For a height function, the gradient points toward the steepest uphill direction, and its magnitude gives the steepest slope.

Quiz questions

1. For a unit vector $\hat{v}$, how is the directional derivative of a differentiable scalar function calculated?Answer: By taking the dot product $\nabla f\cdot\hat{v}$.
2. For spherical coordinates $(r,\theta,\phi)$, what factor multiplies $\partial f/\partial\phi$ in the gradient component along $\hat{e}_\phi$?Answer: $1/(r\sin\theta)$
3. At a regular point of the level surface $F(x,y,z)=c$, how is $\nabla F$ oriented?Answer: Perpendicular to the level surface.
4. If $g$ maps an input vector to a vector and $f$ is scalar-valued, how is the gradient of $f\circ g$ computed?Answer: Apply the transpose Jacobian of $g$ to the outer gradient.
5. What follows from $\nabla f(p)=0$ for a differentiable function at $p$?Answer: The point is stationary, but its type is not determined by the gradient alone.