Objectives

• Understand the directional derivative as the rate of change of \(f(x,y)\) in an arbitrary direction, generalizing the partial derivatives \(f_x\) and \(f_y\text{,}\) which are the special cases in the \(\mathbf i\) and \(\mathbf j\) directions.

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• Interpret the directional derivative geometrically as the slope of the tangent line to the curve in which a vertical plane through \(P_0\text{,}\) in the direction of \(\mathbf u\text{,}\) cuts the surface \(z = f(x,y)\text{.}\)

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• Understand the gradient vector \(\nabla f\) and its relationship to the directional derivative through \(\left(D_{\mathbf u} f\right)_{P_0} = \left(\nabla f\right)_{P_0}\cdot\mathbf u\text{.}\)

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• Interpret the directional derivative through the angle between \(\nabla f\) and \(\mathbf u\text{,}\) and understand why \(f\) changes most rapidly in the direction of the gradient.

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• Understand why the gradient vector is perpendicular to the level curves of \(f\text{.}\)

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