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Given a surface $f(x, y) = z$, a level set for the surface, and a point on that set, how to know that the slope of the tangent line to the level curve and the gradient vector are orthogonal?

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Consider any $C^1$ path $c$, defined on an non-empty interval $J$ such that $c(J)$ is a subset of the domain of the $C^1$-map $f\colon\boldsymbol R^n\to\boldsymbol R$. Then $f\mathop{\circ}c$ is a real-valued $C^1$-map defined on $J$. Its derivative is according to the chain rule $$(f\mathop{\circ}c)'=\langle\nabla (f\mathop{\circ}c),c'\rangle.$$ Now if $c$ is a level curve, $f\mathop{\circ}c$ is constant, hence $(f\mathop{\circ}c)'=0$.

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