Tangent Hyperplane and Optimal Bundle

Instead of lines being tangent, the planes are tangent now. Three-dimensional graph. Curvy graph thing. The plane intersects with the graph thing at one point. Kinda like indifference curve and budget constraint but in 3-dimensional form.

https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/tangent/tangent3.gif

Indifference surface instead of indifference curve. Budget plane instead of budget curve. 3 DIMENSIONS. Budget hyperplane is tangent to the indifference surface at the optimal bundle.

Finding the partial derivatives and putting them in the vector is the same. This is for the Indifference surface.

Make this equal to the vector of the coefficients of the budget plane equation multiplied by lambda(some arbitrary constant).

Find x and y in terms of z or more generally, 2 variables in terms of 1 variable.

The vector extending from the tangent point between the plane and the surface is normal to the plane.

Intuition:

http://math.stackexchange.com/questions/510420/what-is-the-intuition-behind-the-unit-normal-vector-being-the-derivative-of-the

http://mathoverflow.net/questions/1977/why-is-the-gradient-normal