Linear algebra studies vector spaces and linear mappings between those spaces.
What tools do we use for NON-linear mappings between vector spaces?
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The field you are concerned with is called nonlinear functional analysis. (Calculus of variations can be considered to fall in here as well.) There are many tools in nonlinear functional analysis, though they are often fairly specialized since there are so many nonlinear mappings between topological vector spaces.
Common tools involve fixed-point theorems/root-finding theorems (think Newton's method, Brouwer fixed point theorem and its generalizations), generalizations of differentiability, bifurcation theory, and Morse theory/analysis of critical points, and this is very far from an exhaustive list. The tools by nature have to be fairly topological or analytic, because you can no longer take advantage of the algebraic properties a linear mapping enjoys.