Question

Prove that if \(f(\mathbf{r})\) and \(g(\mathbf{r})\) are any two scalar functions of \(\mathbf{r},\) then \(\nabla(f g)=f \nabla g+g \nabla f\)

Short answer

The product rule for differentiation proves that \(\nabla(fg) = f \nabla g + g \nabla f\).

Step-by-step solution

Understand the Gradient of a Product

The task is to show that the gradient of a product of two scalar functions \(f(\mathbf{r})\) and \(g(\mathbf{r})\) follows the product rule. The gradient in vector calculus for a scalar field is the vector of its partial derivatives.

Apply the Gradient Operator to the Product

The gradient of the product \(fg\) is given by \(abla(fg) = \left(\frac{\partial}{\partial x}(fg), \frac{\partial}{\partial y}(fg), \frac{\partial}{\partial z}(fg)\right)\).

Apply Product Rule to Each Partial Derivative

Using the product rule for differentiation, compute the partial derivative of \(fg\) with respect to each variable:- \(\frac{\partial}{\partial x}(fg) = \frac{\partial f}{\partial x} g + f \frac{\partial g}{\partial x}\)- \(\frac{\partial}{\partial y}(fg) = \frac{\partial f}{\partial y} g + f \frac{\partial g}{\partial y}\)- \(\frac{\partial}{\partial z}(fg) = \frac{\partial f}{\partial z} g + f \frac{\partial g}{\partial z}\).

Reconstruct the Gradient Components

Substitute the results from Step 3 into the gradient formula:\[ abla(fg) = \left( \frac{\partial f}{\partial x}g + f\frac{\partial g}{\partial x}, \frac{\partial f}{\partial y}g + f\frac{\partial g}{\partial y}, \frac{\partial f}{\partial z}g + f\frac{\partial g}{\partial z} \right). \]

Factor and Rearrange Components

Factor out the components from the result:\[ abla(fg) = g(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}) + f(\frac{\partial g}{\partial x},\frac{\partial g}{\partial y},\frac{\partial g}{\partial z}) \]This simplifies to:\[ abla(fg) = gabla f + fabla g. \]

Conclusion: Verify Result Matches Desired Expression

The expression \(abla(fg) = f abla g + g abla f\) has been successfully derived using the product rule, thus proving the statement.

Key concepts

Product Rule in Calculus

The product rule in calculus is a fundamental principle used when differentiating the product of two functions. It can be stated simply: to differentiate the product of two functions, apply the derivative to each function in turn, keeping the other constant, and then sum the results. In mathematical notation, for functions \(u(x)\) and \(v(x)\):
\[ (uv)' = u'v + uv' . \]
This rule ensures that the contribution to the rate of change of the product is accounted for by both functions. In vector calculus, we extend this concept to scalar and vector fields, which is especially useful for functions of multiple variables.

Scalar Functions

Scalar functions are mathematical functions that map a vector space to a scalar (a single real number). Unlike vectors, they have no direction. Common examples include temperature, pressure, or density at a point in space.
For a scalar function of a vector \(\mathbf{r} = (x, y, z)\), say \(f(\mathbf{r})\), it assigns a single scalar value to each position \(\mathbf{r}\) in space.

• They are straightforward to visualize because they only involve a scalar value.
• Many physical phenomena can be described by scalar fields, such as sound intensity, gravitational potential, or electrical potential.

Understanding scalar functions is crucial for integrating concepts like the gradient, divergence, or Laplacian in vector calculus, each providing different insights into a scalar field's behavior.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and the operations on these fields. It extends the basic principles of calculus to manipulations involving vectors, making it invaluable in physics and engineering. Some key operations in vector calculus include:

• Gradient: It measures the rate and direction of change in a scalar field. The gradient of a scalar field is itself a vector, pointing in the direction of the greatest rate of increase.
• Divergence: Represents the magnitude of a source or sink at a given point in a vector field.
• Curl: Measures the rotation of a vector field around a point.

Each of these operations serves to analyze different aspects of vector fields, helping to solve real-world problems related to force fields, fluid dynamics, and electromagnetism. Understanding these core operations is essential for navigating topics like the gradient of a product, where vector calculus becomes a powerful tool for deeper analysis.