For ( z = f(x,y) ) with ( x = g(s,t), y = h(s,t) ): [ \frac\partial z\partial s = \frac\partial f\partial x \frac\partial x\partial s + \frac\partial f\partial y \frac\partial y\partial s ] (similar for ( t )). If ( F(x,y,z) = 0 ) defines ( z ) implicitly: [ \frac\partial z\partial x = -\fracF_xF_z, \quad \frac\partial z\partial y = -\fracF_yF_z ] (provided ( F_z \neq 0 )). 12. Optimization (Unconstrained) Find local extrema of ( f: \mathbbR^n \to \mathbbR ).
Solve: [ \nabla f = \lambda \nabla g, \quad g(\mathbfx) = c ] where ( \lambda ) is the Lagrange multiplier. multivariable differential calculus
Existence of all partial derivatives does not guarantee differentiability (continuity of partials does). 7. The Gradient [ \nabla f(\mathbfx) = \left( \frac\partial f\partial x_1, \dots, \frac\partial f\partial x_n \right) ] For ( z = f(x,y) ) with (
