Problem 33 - Thomas Calculus

Question

a. Exact differential form How are the constants \(a, b,\) and \(c\) related if the following differential form is exact? $$\left(a y^{2}+2 c z x\right) d x+y(b x+c z) d y+\left(a y^{2}+c x^{2}\right) d z$$ b. Gradient field For what values of \(b\) and \(c\) will $$\mathbf{F}=\left(y^{2}+2 c z x\right) \mathbf{i}+y(b x+c z) \mathbf{j}+\left(y^{2}+c x^{2}\right) \mathbf{k}$$ be a gradient field?

Step-by-Step Solution

Answer

a. Constants are related by \( 2a = b \) and \( c = 2a \). b. Values for \( b \) and \( c \) are both \( 2a \).

Step 1: Understanding Exact Differential Forms

A differential form \( M(x, y, z)\, dx + N(x, y, z)\, dy + P(x, y, z)\, dz\) is exact if there exists a function \( f(x, y, z) \) such that \( df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz = M dx + N dy + P dz \). For this to be true, certain conditions on the partial derivatives, namely \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \), \( \frac{\partial N}{\partial z} = \frac{\partial P}{\partial y} \), and \( \frac{\partial P}{\partial x} = \frac{\partial M}{\partial z} \), must be satisfied.

Step 2: Applying the Conditions for Exactness

Given \( M = ay^{2} + 2czx \), \( N = y(bx + cz) \), and \( P = ay^{2} + cx^{2} \):- First condition: \( \frac{\partial M}{\partial y} = 2ay = \frac{\partial N}{\partial x} = yb \). This implies \( 2a = b \).- Second condition: \( \frac{\partial N}{\partial z} = yc = \frac{\partial P}{\partial y} = 2ay \). This gives \( c = 2a \).- Third condition is already satisfied as \( \frac{\partial P}{\partial x} = 2cx = \frac{\partial M}{\partial z} = 2cx \), which is true regardless of \(a, b,\) or \(c\).

Step 3: Conclusion for Exactness

The constants are related by: \( 2a = b \) and \( c = 2a \) to ensure the differential form is exact.

Step 4: Understanding Gradient Fields

A vector field \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} + P\mathbf{k} \) is a gradient field if \( \mathbf{F} = abla f \), which implies that the curl of \( \mathbf{F} \) must be zero, i.e., \( abla \times \mathbf{F} = \vec{0} \). This requires computing \( abla \times \mathbf{F} \) and setting each component to zero.

Step 5: Computing the Curl of \( \mathbf{F} \)

The curl of \( \mathbf{F} \) is computed as:\[abla \times \mathbf{F} = \left(\frac{\partial P}{\partial y} - \frac{\partial N}{\partial z}\right) \mathbf{i} - \left(\frac{\partial P}{\partial x} - \frac{\partial M}{\partial z}\right) \mathbf{j} + \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) \mathbf{k}\]Substitute in the given components:\- \( \frac{\partial P}{\partial y} = 2y \) and \( \frac{\partial N}{\partial z} = yc \) give \( 2a = c \).- The \( \mathbf{j} \) component \( \frac{\partial P}{\partial x} = 2cx = \frac{\partial M}{\partial z} = 2cx \), which holds true.- The \( \mathbf{k} \) component \( \frac{\partial N}{\partial x} = yb \) and \( \frac{\partial M}{\partial y} = 2ay \) require \( b = 2a \).

Step 6: Conclusion for Gradient Field

To make \( \mathbf{F} \) a gradient field, choose values such that \( b = 2a \) and \( c = 2a \).

Key Concepts

Gradient Field

A gradient field is a special type of vector field where each point in the field can be represented as the gradient of a scalar function. This means that if we have a vector field \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} + P\mathbf{k} \), it is a gradient field if there exists a function \( f(x, y, z) \) such that \( \mathbf{F} = abla f \). Basically, \( f \) is the potential function, and \( \mathbf{F} \) is the gradient of this potential function.
To determine if a vector field is a gradient field, the curl of the vector field, represented by \( abla \times \mathbf{F} \), must be zero. This is a key property of gradient fields. When \( abla \times \mathbf{F} = \vec{0} \), it implies that the vector field \( \mathbf{F} \) is irrotational, meaning it has no 'twists' or 'circulations.'
In the exercise, the vector field \( \mathbf{F} \) needs to meet certain conditions on its components to be a gradient field, specifically that \( b = 2a \) and \( c = 2a \). This ensures that each component of the curl is zero, confirming that \( \mathbf{F} \) is a gradient field.

Partial Derivatives

Partial derivatives represent the rate of change of a multivariable function with respect to one variable while keeping other variables constant. For a function \( f(x, y, z) \), the partial derivatives are \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \).
In exact differential forms, partial derivatives help verify exactness. Specifically, for exactness, the partial derivatives must satisfy the conditions:

• \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \)
• \( \frac{\partial N}{\partial z} = \frac{\partial P}{\partial y} \)
• \( \frac{\partial P}{\partial x} = \frac{\partial M}{\partial z} \)

These conditions ensure that there exists a scalar function that the differential form can be derived from.
In the exercise, by using partial derivatives, relations between the constants \( a, b, \) and \( c \) were determined, ensuring the differential form is exact. The conditions led to the relationships \( 2a = b \) and \( c = 2a \), showing how partial derivatives are crucial in confirming exactness.

Vector Calculus

Vector calculus is a branch of mathematics dealing with vector fields and operations like differentiation and integration of vector-valued functions. Key operations include the gradient, divergence, and curl. These operations allow mathematicians and scientists to analyze vector fields in different contexts, such as fluid flow, electromagnetism, and more.
In vector calculus, an exact differential form is an equation that can be integrated perfectly to yield a scalar potential function. For example, to establish the exactness of a differential form, one must ensure certain conditions on the partial derivatives. This is a vital step linking it to the concept of a gradient field.
The vector field \( \mathbf{F} \) in the exercise is analyzed using vector calculus techniques, particularly the computation of the curl. The curl is a vector operation that helps determine if a field is conservative (like a gradient field) or not. When the curl is zero, it indicates the absence of rotation and the potential to be expressed as the gradient of a scalar function.

• Gradient: measures the slope of a scalar field.
• Curl: measures the rotation of a vector field.

These concepts together help in solving complex problems involving vector fields and exact differential forms, offering insights into how vector fields behave and can be simplified.