Problem 31 - Calculus Early Transcendentals

Question

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{div}(k \mathbf{F})=k \operatorname{div} \mathbf{F} $$

Step-by-Step Solution

Answer

The identity \( \operatorname{div}(k \mathbf{F}) = k \operatorname{div} \mathbf{F} \) is proven using constant differentiation rules.

Step 1: Understand the Divergence Operator

The divergence of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is given by the operator applied as \( \operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \). It essentially measures how much a vector field is spreading out or converging.

Step 2: Apply Divergence Operator to Scaled Vector

We want to consider the vector field \( k \mathbf{F} = (k F_1, k F_2, k F_3) \). So, apply the divergence operator to this scaled vector: \( \operatorname{div}(k \mathbf{F}) = \frac{\partial (k F_1)}{\partial x} + \frac{\partial (k F_2)}{\partial y} + \frac{\partial (k F_3)}{\partial z} \).

Step 3: Use Constant Rule in Differentiation

Since \( k \) is a constant, it can be factored out of the partial derivatives: \( \operatorname{div}(k \mathbf{F}) = k \frac{\partial F_1}{\partial x} + k \frac{\partial F_2}{\partial y} + k \frac{\partial F_3}{\partial z} \).

Step 4: Compute the Final Expression

Factor \( k \) out of the entire expression: \( \operatorname{div}(k \mathbf{F}) = k \left( \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \right) \). This is equivalent to \( k \operatorname{div} \mathbf{F} \).

Step 5: Conclusion

The identity is proven and can be written as \( \operatorname{div}(k \mathbf{F}) = k \operatorname{div} \mathbf{F} \). This shows that scaling a vector field by a constant scales its divergence by the same constant.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and the differentiation and integration of vectors. It helps in understanding physical concepts such as velocity, force fields, and fluid flow. The key concepts in vector calculus include:

• Vector Fields: Functions that assign a vector to each point in space. For example, the vector field \( \mathbf{F}(x, y, z) \) assigns a vector resulting from combining coordinate functions \( F_1, F_2, \) and \( F_3 \).
• Divergence: This operator measures how a vector field spreads out from a point. Mathematically, it is represented as \( \operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \).
• Curl and Gradient: Other differential operations in vector calculus include curl (which measures rotation) and gradient (which gives the direction of the steepest ascent of a function).

The Divergence Theorem, a central topic in vector calculus, relates the flow (or flux) of a vector field through a surface to the behavior of the field inside the surface. A strong understanding of these operations allows mathematicians and scientists to analyze physical phenomena.

Partial Derivatives

Partial derivatives are a fundamental tool in calculus, specifically useful when dealing with functions of multiple variables. If you have a function \( z = f(x, y) \), the partial derivatives are the rates at which \( z \) changes as \( x \) or \( y \) are changed, while the other variable is held constant.

• Notation: Partial derivatives are denoted using the symbol \(\frac{\partial}{\partial x}\) or \(\frac{\partial}{\partial y}\).
• Constant Factor Rule: If a function is multiplied by a constant, the constant can be factored out of the derivative. For example, \( \frac{\partial}{\partial x}(k \cdot f(x)) = k \cdot \frac{\partial f(x)}{\partial x} \).
• Applications: These derivatives help in constructing tangent planes and in optimizing functions with constraints, which are common in physics and engineering situations.

When looking at divergence, partial derivatives tell us how much the vector component changes in its respective direction. Understanding how these derivatives work is crucial for calculating divergence correctly.

Mathematical Proofs

Mathematical proofs employ logical reasoning to show that a statement is true. In the context of vector calculus, proving identities like \( \operatorname{div}(k \mathbf{F}) = k \operatorname{div} \mathbf{F} \) involves proving that mathematical operations work consistently based on defined rules.

• Format: Mathematical proofs often follow structured formats like direct proof, proof by contradiction, or induction.
• Logical Steps: Start by assuming the premises (what you know), apply logical operations and reach a conclusion.
• Verification: Each step must be justified based on mathematical rules, ensuring the argument is valid.

In our exercise, we justify each step of applying the divergence operator and using the constant rule in differentiation systematically. Developing these skills enhances problem-solving ability and understanding of mathematical concepts.