Normal Property of the Gradient

Proposition (Normal Property of the Gradient) Suppose the function is differentiable at the point and that the gradient at satisfies Then is orthogonal to the level surface (or curve) of through .

    Proof. Let be any smooth curve on the level surface that passes through and describe the curve by the vector function for all in some interval . We will show that the gradient is orthogonal to the tangent vector at Because lies on the level surface, any point on must satisfy and by applying the chain rule, we obtain





Suppose at Then







since  



We also know that for all in Thus, we have



and it follows that We are given that and because the curve is smooth. Therefore, is orthogonal to as required.

Example (Normal Property of the Gradient) Sketch the level curve corresponding to for the function and find a normal vector at the point

    Solution. The level curve for is a hyperbola given by as shown. The gradient vector is perpendicular to the level curve. We have



so at the point is the required normal.

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