Calc 3 - Chapter 12.6 - Directional Derivatives & Gradients

What two things do you need for an equation of a line?

A Point, and a Vector.

Normal Vector: ?

A vector that is perpendicular to a specific Plane. Does not simply mean, "perpendicular".

Formula: Unit Vector, when you are given the angle?

Cosineθ "i" + Sineθ "j". Plug in the given angle and solve.

Formula: Directional Derivative:

Gradient of "F of "x","y", DOTTED with a Unit Vector.

What does the "Max Value of Gradient" mean?

It means it's the maximum value of the slope of the Surface, not just the tangent line. It's when the Unit Vector is going parallel to the Gradient Vector. That happens when Cosine equals 1.

Directional Derivative: What is it, basically?

It's a SCALAR that's the slope of the tangent line, IN THE DIRECTION of a SPECIFIED vector, on that surface, through the given point. Basically, when wanting a vector that matches another vector, but able to use at any point, we create the equation of the vector's slope, then plug in the points given.

Formula: Max Value of Gradient?

It's the Magnitude of the Gradient. ‖Gradient‖

Directional Derivative is a Scalar or Vector?

Scalar.

Formula: Normal Vector of a Plane? (Plane Format)

Simply use the coefficients of the Plane as it's Normal Vector. Example: Plane 2x+4y+2z=18, would have a Normal Vector of: <2 , 4 , 2 >

How to test if Perpendicular?

The Dot Product = 0.

What does "Min Value of Gradient" mean?

The minimum value of slope. It's when Cosine equals -1, aka pie (π). It's negative or opposite of the max Gradient.

If Two Planes are Parallel, then what else can be implied to be parallel as well?

Their Normal Vectors.

Gradient is a Scalar or a Vector?

Vector. It's a vector in the "slope" of a given point. When dotted with a Unit Vector, you get the Directional Derivative.