Problem 65 - Calculus: Early Transcendental Functions

Question

Show that if \(\mathbf{G}=\nabla \times \mathbf{H},\) for some vector field \(\mathbf{H}\) with continuous partial derivatives, then \(\nabla \cdot \mathbf{G}=0\).

Step-by-Step Solution

Answer

Using the vector calculus identity that the divergence of the curl of any vector field is 0, it can be concluded that \(\nabla \cdot \mathbf{G} = 0\).

Step 1: Identification of the given vector field

First, it's recognized that \mathbf{G} is presented as the curl of another vector field \mathbf{H}, i.e. \(\mathbf{G}=\nabla \times \mathbf{H}\). The task is to prove \(\nabla \cdot \mathbf{G}=0\).

Step 2: Finding divergence of the curl

Since divergence is a scalar result of a vector operation, applying divergence to the curl of \mathbf{H} will yield a scalar result. So, the operation \(\nabla \cdot (\nabla \times \mathbf{H})\) is performed, which results in the divergence of the vector field \mathbf{G}.

Step 3: Application of the vector calculus identity

In vector calculus, it's a defined property that the divergence of the curl of any vector field is always zero. This is a well-known identity, i.e. \(\nabla \cdot (\nabla \times \mathbf{H}) = 0\). This identity fulfills the requirement of the problem statement and thus completes the proof.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiation and integration of vector fields. It's crucial for physics and engineering, as it helps describe phenomena like electromagnetic fields and fluid flow.
Vector calculus operates in multiple dimensions and involves using vectors and scalar fields. The curl, divergence, and gradient are key operators in vector calculus, each providing insights into the behavior of vector fields.

• **Curl**: Measures the rotation of a vector field. If the curl is zero, the field is irrotational.
• **Divergence**: Represents the magnitude of a source or sink at a given point in a vector field. For a closed surface, it provides a measure of how much the field is "spreading out".
• **Gradient**: Indicates the direction and rate of fastest increase of a scalar field.

Understanding these concepts aids in solving complex vector calculus problems effectively, as demonstrated in the exercise with the identity involving divergence and curl.

Vector Fields

Vector fields assign a vector to every point in a space, providing direction and magnitude information. They're fundamental to understanding physical phenomena.
Mathematically, a vector field is represented as a function that takes a location as input and outputs a vector. Common examples include magnetic fields and velocity fields in fluid dynamics.

• **Representation**: Generally written as \( \mathbf{F}(x, y, z) = (F_1(x, y, z), F_2(x, y, z), F_3(x, y, z)) \), encompassing three dimensions.
• **Properties**: An important property of vector fields is continuity, meaning the vector elements change smoothly as you move through space.
• **Operations**: Key operations include calculating divergence and curl, which offer insights about flow behavior and rotation.

This exercise demonstrates using vector field operations to derive properties like the divergence of the curl being zero.

Continuous Partial Derivatives

Continuous partial derivatives are a fundamental requirement for analyzing vector fields where smooth changes are critical.
When we say a vector field has continuous partial derivatives, it indicates that each component of the vector field changes smoothly, without abrupt jumps, as the input variables change.

• **Definition**: A partial derivative measures how a function changes as one particular variable changes while keeping others constant.
• **Continuity**: For a function to have continuous partial derivatives, these derivatives themselves must vary smoothly across its domain.
• **Importance**: In the given exercise, continuous partial derivatives ensure that operations like curl and divergence are well-defined and lead to meaningful results.

We need continuous derivatives to apply certain identities and theorems in vector calculus, such as the one used in this solution.

Vector Calculus Identities

Vector calculus identities are equations or rules that hold true for vector fields under certain conditions. They're vital for simplifying complex vector operations.
These identities help in proving properties of vector fields. For instance, the exercise relies on the identity that the divergence of the curl is zero, which is universally accepted in mathematics.

• **Common Identities**: Include \( abla \cdot (abla \times \mathbf{H}) = 0 \) and \( abla \times (abla f) = 0 \) for any scalar function \( f \).
• **Purpose**: These identities help verify fundamental properties of vector fields and simplify derivations in theoretical and applied contexts.
• **Application**: Understanding these identities enhances problem-solving skills in vector calculus, making it possible to derive and prove new relationships between vector fields.

In this example, the identity discussed is crucial in establishing the behavior of the curl operation, strengthening understanding of vector field operations.