In civil engineering and hydrology, understanding the flow of water across a landscape is critical for designing drainage systems. A topographic map can be modeled as a multivariable function $z = E(x, y)$, where $z$ represents the elevation at any longitude $x$ and latitude $y$. The Gradient Vector $\nabla E = \langle \frac{\partial E}{\partial x}, \frac{\partial E}{\partial y} \rangle$ mathematically defines the direction of steepest ascent at any specific point. Conversely, water will naturally flow in the direction of steepest descent, which is the negative gradient ($-\nabla E$). Furthermore, the contour lines on the map (lines of constant elevation) are orthogonal (perpendicular) to the gradient vector at every point.

An aerospace engineer is designing a pressurized cylindrical fuel tank with hemispherical ends. The tank must hold a specific volume $V_0$ (the constraint equation $g(r, h) = V_0$). However, the materials used for the cylindrical body and the hemispherical ends have different costs per square meter. The engineer wants to minimize the total material cost function $C(r, h)$. This is a classic constrained optimization problem perfectly suited for the Method of Lagrange Multipliers. By setting $\nabla C = \lambda \nabla g$, the engineer can find the optimal radius $r$ and cylinder length $h$ that minimize the cost while still meeting the exact volume requirement.

Find the partial derivatives $f_x$ and $f_y$ for the function: $f(x, y) = 3x^2y - 5xy^3 + 2x - 7$.

Find $f_x$ and $f_y$ for the transcendental function $f(x, y) = e^{xy} \sin(x^2 y)$.

Verify Clairaut's Theorem ($f_{xy} = f_{yx}$) for the function $f(x, y) = x^3 y^2 + \cos(xy)$.

Find the gradient vector for the function $f(x, y) = x^2 y^3 - 4y$ at the point $(2, -1)$, and determine the maximum rate of increase at that point.

Find the directional derivative of $f(x, y) = x^2e^y$ at the point $(2, 0)$ in the direction of the vector $\mathbf{v} = \langle 3, 4 \rangle$.

At the point $(1, 1)$, in what direction does the function $f(x, y) = x^2 + 3xy - 2y^2$ decrease the fastest?

Find and classify all critical points of the function $f(x,y) = 2x^2 + y^2 + 8x - 6y + 20$.

Find and classify the critical points of $f(x, y) = x^3 - 3x + 3xy^2$.

Find the maximum and minimum values of $f(x,y) = xy$ subject to the constraint $x^2 + y^2 = 8$.

Let $z = x^2 y$. Find the total derivative $\frac{dz}{dt}$ if $x = t^2$ and $y = 3t$.

The dimensions of a closed rectangular box are measured as $80\text{ cm}, 60\text{ cm}$, and $50\text{ cm}$, with a possible error of $0.2\text{ cm}$ in each dimension. Use the total differential to estimate the maximum error in calculating the volume of the box.