Mathematics · University
Gradients, tangent planes and extrema · revision notes
The gradient collects first partial derivatives and determines directional derivatives through a dot product.
Key ideas and conditions
Gradient
The gradient collects first partial derivatives and determines directional derivatives through a dot product.
Tangent plane
Linearization at a differentiable point uses f(a,b)+fₓ(a,b)(x−a)+fᵧ(a,b)(y−b).
Stationary points
A zero gradient is necessary for an interior differentiable extremum, not sufficient.
Hessian test
In two variables, positive determinant and positive fₓₓ identify a strict local minimum; negative determinant identifies a saddle.
Explain without looking
- If a stationary point has Hessian determinant zero, what does the ordinary second-derivative test conclude?
- Give a differentiable function whose gradient is zero at the origin but that has no local extremum there.
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.