The directional derivative of the scalar field $\phi(x, y, z) = x^2 y + y^2 z + z^2 x$ at the point $P(1, 2, -1)$ in the direction of the unit vector pointing toward the point $Q(3, 0, 0)$ is:

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### 1. Strategy & Theoretical Principles The directional derivative of a scalar field $\phi$ at a point $P$ along a direction $\vec{u}$ is: $$D_{\vec{u}} \phi = \nabla \phi(P) \cdot \hat{u}$$ where $\hat{u} = \frac{\vec{PQ}}{\|\vec{PQ}\|}$ is the unit vector along the directed line segment from $P$ to $Q$. ### 2. Analytical Formulation 1. Gradient of $\phi = x^2 y + y^2 z + z^2 x$: $$\nabla \phi = \left(2xy + z^2\right) \hat{i} + \left(x^2 + 2yz\right) \hat{j} + \left(y^2 + 2zx\right) \hat{k}$$ 2. Evaluate $\nabla \phi$ at $P(1, 2, -1)$: $$\frac{\partial \phi}{\partial x} = 2(1)(2) + (-1)^2 = 4 + 1 = 5$$ $$\frac{\partial \phi}{\partial y} = (1)^2 + 2(2)(-1) = 1 - 4 = -3$$ $$\frac{\partial \phi}{\partial z} = (2)^2 + 2(-1)(1) = 4 - 2 = 2$$ $$\nabla \phi(1, 2, -1) = 5 \hat{i} - 3 \hat{j} + 2 \hat{k}$$ ### 3. Step-by-Step Calculation Vector $\vec{PQ}$ from $P(1, 2, -1)$ to $Q(3, 0, 0)$: $$\vec{PQ} = (3 - 1) \hat{i} + (0 - 2) \hat{j} + (0 - (-1)) \hat{k} = 2 \hat{i} - 2 \hat{j} + 1 \hat{k}$$ Magnitude: $$\|\vec{PQ}\| = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$$ Unit vector $\hat{u}$: $$\hat{u} = \frac{2}{3} \hat{i} - \frac{2}{3} \hat{j} + \frac{1}{3} \hat{k}$$ Evaluating the dot product: $$D_{\vec{u}} \phi = (5)\left(\frac{2}{3}\right) + (-3)\left(-\frac{2}{3}\right) + (2)\left(\frac{1}{3}\right) = \frac{10 + 6 + 2}{3} = \frac{18}{3} = 6.0$$ *Correction in manual check*: $10 + 6 + 2 = 18 \implies 18/3 = 6.0$! Let's set the correct answer to 6.0 (tolerance 5.95 to 6.05). ### 4. Distractor Forensics - Forgetting to normalize vector $\vec{PQ}$ gives $18.0$. - Sign error on $\partial \phi / \partial y$ gives $(10 - 6 + 2)/3 = 2.0$. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always divide $\vec{PQ}$ by its magnitude to form a *unit* direction vector $\hat{u}$. - **Standard Reference:** Kreyszig, E., *Advanced Engineering Mathematics*, Chapter 9 (Vector Differential Calculus).

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