Chapter 4: Problem 14
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
Expert verified
The product rule for differentiation proves that \(\nabla(fg) = f \nabla g + g \nabla f\).
Step by step solution
01
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.
02
Apply the Gradient Operator to the Product
The gradient of the product \(fg\) is given by \(abla(fg) = \left(\frac{\partial}{\partial x}(fg), \frac{\partial}{\partial y}(fg), \frac{\partial}{\partial z}(fg)\right)\).
03
Apply Product Rule to Each Partial Derivative
Using the product rule for differentiation, compute the partial derivative of \(fg\) with respect to each variable:- \(\frac{\partial}{\partial x}(fg) = \frac{\partial f}{\partial x} g + f \frac{\partial g}{\partial x}\)- \(\frac{\partial}{\partial y}(fg) = \frac{\partial f}{\partial y} g + f \frac{\partial g}{\partial y}\)- \(\frac{\partial}{\partial z}(fg) = \frac{\partial f}{\partial z} g + f \frac{\partial g}{\partial z}\).
04
Reconstruct the Gradient Components
Substitute the results from Step 3 into the gradient formula:\[ abla(fg) = \left( \frac{\partial f}{\partial x}g + f\frac{\partial g}{\partial x}, \frac{\partial f}{\partial y}g + f\frac{\partial g}{\partial y}, \frac{\partial f}{\partial z}g + f\frac{\partial g}{\partial z} \right). \]
05
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:\[ abla(fg) = gabla f + fabla g. \]
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
\[ (uv)' = u'v + uv' . \]
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.
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.
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.