91Ó°ÊÓ

The gradient vector is a fundamental concept in multivariable calculus. It tells us the direction of the steepest ascent of a function at a given point. For a function of three variables like \(f(x, y, z)\), the gradient \(abla f\) is formed by the partial derivatives of \(f\) with respect to each variable.

• The x-component is \(\frac{\partial f}{\partial x}\)
• The y-component is \(\frac{\partial f}{\partial y}\)
• The z-component is \(\frac{\partial f}{\partial z}\)

By calculating these, we obtain a vector, \((\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})\), which forms the gradient vector \(abla f\). For the function \(f(x, y, z) = xy + yz + zx\), the gradient is \(abla f = (y + z, x + z, x + y)\). This vector provides crucial information about how the function changes at any point \((x, y, z)\).

Partial derivatives allow us to understand how a function changes with respect to one variable, keeping the others constant. Imagine adjusting one knob on a machine while the other knobs stay still; this is analogous to calculating a partial derivative.For the given function \(f(x, y, z) = xy + yz + zx\):

• \(\frac{\partial f}{\partial x} = y + z\): This shows how \(f\) changes if only \(x\) changes.
• \(\frac{\partial f}{\partial y} = x + z\): This represents the change in \(f\) if only \(y\) varies.
• \(\frac{\partial f}{\partial z} = x + y\): This reflects the variation in \(f\) as \(z\) alone changes.

These expressions are derived by differentiating the function with respect to each variable independently. They offer a way to dissect the behavior of multivariable functions.

A unit vector is a vector with a length (or magnitude) of 1. It retains only the direction of the original vector, with its original magnitude scaled down.To find the unit vector \(\hat{\mathbf{u}}\) in the direction of a given vector \(\mathbf{u}\), we use the formula:\[ \hat{\mathbf{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|} \]where \(\|\mathbf{u}\|\) denotes the magnitude of \(\mathbf{u}\).For \(\mathbf{u} = 3\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}\), its magnitude is \(\sqrt{3^2 + 6^2 + (-2)^2} = 7\). Thus, the unit vector is:\[ \hat{\mathbf{u}} = \left(\frac{3}{7}, \frac{6}{7}, \frac{-2}{7}\right) \]This simplifies the process of finding directional derivatives, as it ensures the direction vector does not impose additional scaling.

The dot product is a method of multiplying two vectors, resulting in a scalar. It encapsulates the idea of projecting one vector onto another.To calculate the dot product of two vectors \(\mathbf{A} = (A_1, A_2, A_3)\) and \(\mathbf{B} = (B_1, B_2, B_3)\), we use the formula:\[ \mathbf{A} \cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3 \]In the context of directional derivatives, the dot product is used to measure how much of the gradient vector \(abla f\) projects onto the given unit direction vector \(\hat{\mathbf{u}}\).For our problem, with \(abla f = (1, 3, 0)\) and \(\hat{\mathbf{u}} = \left(\frac{3}{7}, \frac{6}{7}, \frac{-2}{7}\right)\), the dot product is:\[ D_{\mathbf{u}}f = 1 \times \frac{3}{7} + 3 \times \frac{6}{7} + 0 \times \frac{-2}{7} = 3 \]Hence, the directional derivative quantifies the rate of change in a specified direction.