To find the gradient vector field $\nabla f$ of $f(x, y, z) = x^3 y e^{y/z}$, we calculate the partial derivatives of $f$ with respect to $x$, $y$, and $z$.

Step 1: Partial derivative with respect to $x$

$f(x, y, z) = x^3 y e^{y/z}$ Differentiating with respect to $x$, treating $y$ and $z$ as constants: $\frac{\partial f}{\partial x} = 3x^2 y e^{y/z}$

Step 2: Partial derivative with respect to $y$

Now, differentiating $f$ with respect to $y$, treating $x$ and $z$ as constants: $\frac{\partial f}{\partial y} = x^3 e^{y/z} + x^3 y \cdot e^{y/z} \cdot \frac{1}{z} = x^3 e^{y/z} \left(1 + \frac{y}{z}\right)$

Step 3: Partial derivative with respect to $z$

Now, differentiating $f$ with respect to $z$, treating $x$ and $y$ as constants: $\frac{\partial f}{\partial z} = x^3 y \cdot e^{y/z} \cdot \left(-\frac{y}{z^2}\right) = -\frac{x^3 y^2}{z^2} e^{y/z}$

Step 4: Combine the partial derivatives

The gradient vector field $\nabla f$ is: $\nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$ Substituting the partial derivatives: $\nabla f(x, y, z) = \left( 3x^2 y e^{y/z}, \; x^3 e^{y/z} \left(1 + \frac{y}{z}\right), \; -\frac{x^3 y^2}{z^2} e^{y/z} \right)$

Final Answer:

$\nabla f(x, y, z) = \left( 3x^2 y e^{y/z}, \; x^3 e^{y/z} \left(1 + \frac{y}{z}\right), \; -\frac{x^3 y^2}{z^2} e^{y/z} \right)$

Would you like more details on each step or have any questions?

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Here are 5 questions related to this topic for further exploration:

1. How does the gradient vector field change if we alter the function $f(x, y, z)$ slightly, such as by adding a constant?
2. What is the physical interpretation of the gradient vector field in three-dimensional space?
3. How can we find the direction in which $f(x, y, z)$ increases most rapidly at a given point?
4. What would be the divergence of the gradient vector field $\nabla f$?
5. How does the choice of coordinate system (like cylindrical or spherical) affect the calculation of the gradient?

Tip: When calculating gradients, keep track of each variable's role carefully, as treating one incorrectly can lead to mistakes in partial derivatives.