Problem 18 - Calculus

Question

Suppose curl \(\mathbf{F}=0\) and div \(\mathbf{F}=0 .\) (a) Why is \(\mathbf{F}\) the gradient of a potential? (b) Why does the potential satisfy Laplace's equation \(f_{x x}+f_{y y}+f_{z z}=0^{2}\)

Step-by-Step Solution

Answer

(a) \(\mathbf{F}\) is the gradient of a potential because curl is zero. (b) The potential satisfies Laplace's equation because divergence is zero.

Step 1: Understand the Condition curl 𝐹 = 0

The condition \(abla \times \mathbf{F} = 0\) indicates that the vector field \(\mathbf{F}\) is irrotational. This means that \(\mathbf{F}\) has no circulation and is locally the gradient of some scalar function. Thus, \(\mathbf{F}\) can be expressed as \(abla f\), where \(f\) is a scalar potential function.

Step 2: Interpret div 𝐹 = 0

The condition \(abla \cdot \mathbf{F} = 0\) implies that the vector field \(\mathbf{F}\) is divergence-free. This signifies that there is no net flow out of any region in the field, consistent with \(\mathbf{F}\) being expressible as the curl of another vector field.

Step 3: Show 𝐹 as Gradient of a Potential Function

Since \(\mathbf{F}\) is irrotational (curl-free), it can be expressed as the gradient of a potential function \(f\) such that \(\mathbf{F} = abla f\). This is directly derived from the fundamental theorem for gradients, which states a vector field is the gradient of a scalar potential if and only if it is irrotational.

Step 4: Explain Why Potential Satisfies Laplace's Equation

Given \(\mathbf{F} = abla f\) and \(abla \cdot \mathbf{F} = 0\), substitute to get \(abla \cdot abla f = 0\). This equation is equivalent to the Laplace’s equation \(abla^2 f = 0\), meaning the potential function \(f\) must satisfy Laplace's equation.

Key Concepts

Curl

In vector calculus, the concept of _curl_ helps us understand the rotational aspects of a vector field. A vector field, denoted as \( \mathbf{F} \), is often seen as an arrow field representing velocities or forces.
The curl operation shows how much, and in what direction, the vector field rotates around a point. Mathematically, the curl is expressed as \( abla \times \mathbf{F} \).

• If the result is zero, \( abla \times \mathbf{F} = 0 \), it indicates an _irrotational_ field. This means there is no rotation or vortex-like structure within the field.
• In our exercise, since \( \text{curl} \, \mathbf{F} = 0 \), it tells us that \( \mathbf{F} \) is irrotational, meaning \( \mathbf{F} \) can be expressed as the gradient of some scalar potential function.

Divergence

Another fundamental concept in vector fields is _divergence_, captured by the operation \( abla \cdot \mathbf{F} \). This measures the magnitude of a source or sink at a given point. In simpler terms, divergence denotes how much a vector field spreads out or converges at that point.

• If \( abla \cdot \mathbf{F} = 0 \), the field is _divergence-free_. This implies there is no local "outflow" or "inflow" at any given point, similar to an incompressible fluid with constant density.
• Our exercise states \( \text{div} \, \mathbf{F} = 0 \), suggesting no net flow is exiting or entering any region in the field.

Divergence-free fields are special because they can be represented as the curl of another vector field. However, in this case, \( \mathbf{F} \) itself is expressed as the gradient of a potential function due to being irrotational.

Potential Function

A _potential function_, commonly denoted \( f \), is a scalar field from which a vector field can be derived. When a vector field \( \mathbf{F} \) is known to have zero curl, it is often the gradient of such a potential function.

• The gradient of \( f \), written as \( abla f \), produces the vector field \( \mathbf{F} \) in this situation.
• The essence lies in the fundamental theorem for gradients, which tells us a vector field is irrotational if and only if it is the gradient of some scalar potential. This links the concept of a potential function directly to the idea of an irrotational vector field.

In our exercise, since \( \text{curl} \, \mathbf{F} = 0 \), we confirm that \( \mathbf{F} \) is indeed the gradient of a potential function \( f \). This relationship helps simplify the problem into more manageable calculations, easing the study of the vector field's behavior.

Laplace's Equation

_Laplace's Equation_ is a critical equation in mathematical physics, expressed as \( abla^2 f = 0 \). It represents a harmonic function, typically implying that the function \( f \) does not change "on average" around any small region.

• In our scenario, because \( abla \cdot \mathbf{F} = 0 \) and \( \mathbf{F} = abla f \), it simplifies to \( abla \cdot abla f = abla^2 f = 0 \).
• Thus, the potential function \( f \), derived from \( \mathbf{F} \), satisfies Laplace's equation, ensuring it is _harmonic_.

Harmonic functions are very smooth and quite predictable. The satisfaction of Laplace's equation by \( f \) indicates equilibrium and stability, often modeling phenomena like electric and gravitational potential fields in physics.