Problem 4 - Thomas Calculus

Question

Find the gradient fields of the functions $$ g(x, y, z)=x y+y z+x z $$

Step-by-Step Solution

Answer

The gradient field is \((y + z, x + z, y + x)\).

Step 1: Understand the Gradient Concept

The gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function. It is obtained by taking the partial derivatives of the function with respect to each independent variable. For a function \(g(x, y, z)\), the gradient \(abla g\) is given by \(\left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \frac{\partial g}{\partial z} \right)\).

Step 2: Compute Partial Derivative with Respect to x

To find \(\frac{\partial g}{\partial x}\), differentiate the function \(g(x, y, z) = xy + yz + xz\) with respect to \(x\), treating \(y\) and \(z\) as constants. This yields \(\frac{\partial g}{\partial x} = y + z\).

Step 3: Compute Partial Derivative with Respect to y

For \(\frac{\partial g}{\partial y}\), differentiate \(g(x, y, z)\) with respect to \(y\), treating \(x\) and \(z\) as constants. This gives \(\frac{\partial g}{\partial y} = x + z\).

Step 4: Compute Partial Derivative with Respect to z

To find \(\frac{\partial g}{\partial z}\), differentiate \(g(x, y, z)\) with respect to \(z\), keeping \(x\) and \(y\) as constants. This results in \(\frac{\partial g}{\partial z} = y + x\).

Step 5: Formulate the Gradient Vector

Combine the partial derivatives from Steps 2, 3, and 4 to express the gradient field as a vector. Hence, \(abla g = (y + z, x + z, y + x)\).

Key Concepts

Partial Derivatives

Partial derivatives are a powerful tool in calculus, particularly when dealing with functions of multiple variables. When you have a function like \( g(x, y, z) = xy + yz + xz \), the function depends on more than one variable. A partial derivative focuses on how the function changes as one of these variables changes, while the others are held constant.

Here's how partial derivatives work:

• For \( \frac{\partial g}{\partial x} \), you treat \( y \) and \( z \) as constants and differentiate with respect to \( x \).
• For \( \frac{\partial g}{\partial y} \), \( x \) and \( z \) are constants, and you differentiate with respect to \( y \).
• For \( \frac{\partial g}{\partial z} \), \( x \) and \( y \) remain constant, and differentiation happens with respect to \( z \).

This method creates derivatives that reveal the rate of change of the function along each variable's direction. Partial derivatives are foundational for gradient calculations, which we'll explore next.

Vector Fields

When you calculate the gradient of a function, you're creating a vector field. A vector field assigns a vector to every point in space. For functions of multiple variables, the gradient vector field indicates the direction and rate of the steepest increase in the function's value.

In the context of the gradient of \( g(x, y, z) = xy + yz + xz \), the gradient is given by \( abla g = (y + z, x + z, y + x) \). Each component of this gradient vector is the partial derivative of the function with respect to each variable:

• The \( x \)-component, \( y + z \), shows how the function changes as \( x \) varies.
• The \( y \)-component, \( x + z \), indicates the change with variations in \( y \).
• The \( z \)-component, \( y + x \), describes the change with respect to \( z \).

Each point \((x, y, z)\) in space is associated with one such vector, visualizing how the function behaves in three-dimensional space. These insights are crucial for analyzing fluid flows, electromagnetic fields, and other physical processes.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with more than one variable. This field is crucial for many scientific and engineering applications, helping solve complex problems that involve several changing factors.

In multivariable calculus:

• Functions depend on multiple independent variables, such as \( x, y, \text{ and } z \).
• Calculating gradients, like \( abla g = (y + z, x + z, y + x) \), involves determining how functions change along different axes in a multi-dimensional space.
• It's essential for optimization, finding maxima and minima in complex models.

The tools and concepts in multivariable calculus, such as partial derivatives and vector fields, allow for analysis and problem-solving in multiple dimensions, making it indispensable for modern technical fields.