Question

Write the gradients of the following functions: (a) \(f(x, y, z)=x^{2}+y^{3}+z^{4}\) (b) \(f(x, y, z)=x y z\)

Short answer

(a) \(\nabla f = (2x, 3y^2, 4z^3)\); (b) \(\nabla f = (yz, xz, xy)\)."

Step-by-step solution

Understand the Concept of Gradient

The gradient of a scalar function is a vector of its partial derivatives. For a function of variables \(x, y, z\), the gradient is denoted as \(abla f\) and is given by \(\left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\). This vector points in the direction of the greatest rate of increase of the function.

Find Partial Derivative with Respect to x for (a)

For the function \(f(x, y, z) = x^2 + y^3 + z^4\), calculate the partial derivative with respect to \(x\): \(\frac{\partial f}{\partial x} = 2x\).

Find Partial Derivative with Respect to y for (a)

For \(f(x, y, z) = x^2 + y^3 + z^4\), calculate the partial derivative with respect to \(y\): \(\frac{\partial f}{\partial y} = 3y^2\).

Find Partial Derivative with Respect to z for (a)

For \(f(x, y, z) = x^2 + y^3 + z^4\), calculate the partial derivative with respect to \(z\): \(\frac{\partial f}{\partial z} = 4z^3\).

Write the Gradient for Function (a)

The gradient \(abla f\) for the function \(f(x, y, z) = x^2 + y^3 + z^4\) is \(abla f = (2x, 3y^2, 4z^3)\).

Find Partial Derivative with Respect to x for (b)

For the function \(f(x, y, z) = xyz\), calculate the partial derivative with respect to \(x\): \(\frac{\partial f}{\partial x} = yz\).

Find Partial Derivative with Respect to y for (b)

For \(f(x, y, z) = xyz\), calculate the partial derivative with respect to \(y\): \(\frac{\partial f}{\partial y} = xz\).

Find Partial Derivative with Respect to z for (b)

For \(f(x, y, z) = xyz\), calculate the partial derivative with respect to \(z\): \(\frac{\partial f}{\partial z} = xy\).

Write the Gradient for Function (b)

The gradient \(abla f\) for the function \(f(x, y, z) = xyz\) is \(abla f = (yz, xz, xy)\).

Key concepts

Partial Derivatives

A partial derivative represents the derivative of a multi-variable function with respect to one variable, keeping the other variables constant.
When dealing with functions of multiple variables, such as \( f(x, y, z) \), it's important to understand how each variable individually affects the function.
This is where partial derivatives come into play. Partial Derivatives in Function (a):

• For \( f(x, y, z) = x^2 + y^3 + z^4 \), each variable impacts the function differently due to their respective powers.
  The partial derivative with respect to \( x \) is found by treating \( y \) and \( z \) as constants and calculating \( \frac{\partial f}{\partial x} = 2x \).
• Similarly, maintaining \( x \) and \( z \) constant, \( \frac{\partial f}{\partial y} = 3y^2 \).
• Finally, keeping \( x \) and \( y \) constant, we find \( \frac{\partial f}{\partial z} = 4z^3 \).

Partial Derivatives in Function (b):

• In \( f(x, y, z) = xyz \), each variable participates symmetrically, leading to simpler derivatives.
  Here, \( \frac{\partial f}{\partial x} = yz \) when holding \( y \) and \( z \) constant.
• The symmetry continues with \( \frac{\partial f}{\partial y} = xz \), holding \( x \) and \( z \) constant.
• And finally, \( \frac{\partial f}{\partial z} = xy \), keeping \( x \) and \( y \) constant.

This concept of "influence" or "sensitivity" towards a specific variable is crucial when navigating through multi-variable calculus.

Scalar Functions

A scalar function is a mathematical expression that associates a single scalar value to each point in space.
Scalar functions are fundamental as they help describe various physical phenomena like temperature, pressure, or density.
These functions depend on multiple variables and are vital in introducing vector calculus concepts.Given the function \( f(x, y, z) \), each unique combination of \( x, y, \) and \( z \) gives a distinct scalar output.
This can be thought of as assigning a single value, accessible in any location defined by \( (x, y, z) \) coordinates.
In Functions (a) and (b), these scalar outputs change depending on the variables' state.Examples from the Exercise:

• In \( f(x, y, z) = x^2 + y^3 + z^4 \), geometric interpretations can be visualized in 3D space, whereby hills or valleys represent scalar values at each point.
• Function \( f(x, y, z) = xyz \) is another type of scalar function where each variable makes a linear contribution to the scalar result, exhibiting a unique cross-product-like combination of \( x, y, \) and \( z \).

By understanding scalar functions, students are better prepared to analyze how changes in variables influence the overall value and behavior of complex systems.

Directional Derivatives

In calculus, a directional derivative measures the rate at which a function changes at a point in a specified direction.
It's the ultimate tool to understand how steeply a function rises or falls moving in any given direction.
Unlike partial derivatives, which only consider change along coordinate axes, directional derivatives dive deeper into the function’s geometry.Calculating Directional Derivatives:

• To compute a directional derivative, take the gradient of the function and the direction vector.
• Assure the direction vector is normalized (i.e., it has a length of 1).
• The directional derivative is the dot product of the gradient vector and the normalized direction vector.

For example, when assessing a surface described by \( f(x, y, z) \), you’d want to know how the function behaves along vectors that aren’t naturally aligned with the axes.
The gradient helps us extrapolate such behavior efficiently.Connecting to the Gradient:The gradient vector \( abla f \) for \( f(x, y, z) = x^2 + y^3 + z^4 \) was expressed in the exercise as \( (2x, 3y^2, 4z^3) \); this indicates the direction of steepest ascent.
For any other direction, a corresponding directional derivative provides information on how the function changes. Grasping this concept expands our capacity to predict directional behavior, which is essential for optimizations and understanding multidimensional changes.