Gradient and Linearization Concepts

A set of flashcards covering key concepts related to gradients, linearization, and multi-variable calculus, including definitions and formulas.

Gradient (∇f)

The gradient is a vector that contains all the partial derivatives of a function.

|∇f|

The magnitude of the gradient indicates the maximum rate of change of the function at that point.

Directional derivative (D_u f)

A measure of how a function changes as you move in a specific direction, given by the formula D_u f = ∇f · û.

Unit vector (û)

A vector of length 1 in the direction of vector u, calculated as û = u / |u|.

Magnitude of vector |u|

The length of vector u, calculated as |u| = √(a² + b²) for u = ⟨a,b⟩.

Maximum directional derivative value (Max D_u f)

The maximum value of the directional derivative, equal to |∇f|.

Minimum directional derivative value (Min D_u f)

The minimum value of the directional derivative, equal to -|∇f|.

Directional derivative is zero

Occurs when the direction vector is perpendicular to ∇f.

Tangent plane formula (explicit surface)

The equation for the tangent plane given by z − z₀ = fx(x₀,y₀)(x − x₀) + fy(x₀,y₀)(y − y₀).

Implicit surface tangent plane formula

The equation representing the tangent plane for an implicit surface, given by Fx(x − x₀) + Fy(y − y₀) + F_z(z − z₀) = 0.

Normal line formula

The parametric equations for the normal line, given by (x,y,z) = (x₀,y₀,z₀) + t∇F.

Linearization meaning

Linearization approximates a function near a point using the tangent plane.

Linearization formula (2 variables)

L(x,y) = f(x₀,y₀) + fx(x₀,y₀)(x − x₀) + fy(x₀,y₀)(y − y₀).

Linearization formula (3 variables)

L(x,y,z) = f(P) + fx(P)(x − x₀) + fy(P)(y − y₀) + f_z(P)(z − z₀).

Error bound formula for linearization

|E(x,y)| ≤ (M/2)(|x − x₀| + |y − y₀|)².

Meaning of M in error bound

M is the maximum value of the second derivatives in the region.

Critical point definition

A point where the first partial derivatives fx and fy are both zero or undefined.

Finding critical points

Solve the equations fx = 0 and fy = 0 simultaneously.

Second derivative test discriminant (D)

D is calculated as D = fxx fyy − (f_xy)².

Condition for local minimum

To have a local minimum, D > 0 and f_xx > 0.

Condition for local maximum

To have a local maximum, D > 0 and f_xx < 0.

Condition for saddle point

Occurs when D < 0.

When second derivative test fails

The test fails when D = 0.

Tangent line to level curve formula

fx(x₀,y₀)(x − x₀) + fy(x₀,y₀)(y − y₀) = 0.