Word Count : 339

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We are given: * Scalar field: $$ \phi(x, y, z) = \ln(x^2 + y^2 + z^2) $$ * Direction vector: $$ \vec{v} = \hat{i} + 2\hat{j} - \hat{k} $$ * Point of interest: $$ (1, -1, 2) $$ --- ### Step 1: Gradient of φ We compute the gradient of $\phi$: $$ \vec{\nabla} \phi = \frac{\partial \phi}{\partial x} \hat{i} + \frac{\partial \phi}{\partial y} \hat{j} + \frac{\partial \phi}{\partial z} \hat{k} $$ We use the chain rule: $$ \phi = \ln(x^2 + y^2 + z^2) \Rightarrow _________ ____ _________ ______ _____.
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