Problem 34 - 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. $$ \operatorname{curl}(\mathbf{F}+\mathbf{G})=\operatorname{curl} \mathbf{F}+\operatorname{curl} \mathbf{G} $$

Step-by-Step Solution

Answer

The curl of the sum is the sum of the curls: \(\operatorname{curl}(\mathbf{F} + \mathbf{G}) = \operatorname{curl}\mathbf{F} + \operatorname{curl}\mathbf{G}\).

Step 1: Stating the Problem

We need to prove the identity \(\operatorname{curl}(\mathbf{F}+\mathbf{G})=\operatorname{curl}\mathbf{F}+\operatorname{curl}\mathbf{G}\) for vector fields \(\mathbf{F}\) and \(\mathbf{G}\), assuming all required derivatives exist and are continuous.

Step 2: Definition of Curl

The curl of a vector field \(\mathbf{F} = \langle F_1, F_2, F_3 \rangle\) is given by \(\operatorname{curl}\mathbf{F} = abla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right)\).

Step 3: Defining the Curl for \(\mathbf{F} + \mathbf{G}\)

Let \(\mathbf{G} = \langle G_1, G_2, G_3 \rangle\). Then, the curl of the sum \(\mathbf{F} + \mathbf{G} = \langle F_1 + G_1, F_2 + G_2, F_3 + G_3 \rangle\) is given by the operator \(abla \times (\mathbf{F} + \mathbf{G})\).

Step 4: Applying the Curl Operator

Calculate each component:1. \( \frac{\partial (F_3 + G_3)}{\partial y} - \frac{\partial (F_2 + G_2)}{\partial z} = \left(\frac{\partial F_3}{\partial y} + \frac{\partial G_3}{\partial y}\right) - \left(\frac{\partial F_2}{\partial z} + \frac{\partial G_2}{\partial z}\right) \).2. \( \frac{\partial (F_1 + G_1)}{\partial z} - \frac{\partial (F_3 + G_3)}{\partial x} = \left(\frac{\partial F_1}{\partial z} + \frac{\partial G_1}{\partial z}\right) - \left(\frac{\partial F_3}{\partial x} + \frac{\partial G_3}{\partial x}\right) \).3. \( \frac{\partial (F_2 + G_2)}{\partial x} - \frac{\partial (F_1 + G_1)}{\partial y} = \left(\frac{\partial F_2}{\partial x} + \frac{\partial G_2}{\partial x}\right) - \left(\frac{\partial F_1}{\partial y} + \frac{\partial G_1}{\partial y}\right) \).

Step 5: Concluding the Proof

The components from the previous step give us the final curl vector:\[ abla \times (\mathbf{F} + \mathbf{G}) = \left(\operatorname{curl}\mathbf{F} + \operatorname{curl}\mathbf{G}\right) \]Therefore, by linearity, we have shown \(\operatorname{curl}(\mathbf{F}+\mathbf{G}) = \operatorname{curl}\mathbf{F} + \operatorname{curl}\mathbf{G}\).

Key Concepts

Curl of a Vector Field

The concept of the curl of a vector field is tied closely to rotation within a three-dimensional space. Simply put, the curl measures how much a vector field tends to rotate around a point.
It gives a vector result that provides information about the tendency of fluid elements to rotate at a point in the field. The formula for the curl of a vector field \( \mathbf{F} = \langle F_1, F_2, F_3 \rangle \) is:
\[\operatorname{curl}\mathbf{F} = abla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right).\]
If this formula results in a zero vector, then the vector field is said to be irrotational. It essentially means that locally at any point, the vector field has no tendency to rotate around that point.
Understanding the curl can be important in fields like electromagnetism and fluid dynamics, where concepts of field rotation play a crucial role.

Vector Fields

A vector field is a function that assigns a vector to every point in space. Imagine assigning a little arrow at every point in a room.
This arrow could represent the speed and direction of wind at every point or the force field in a region of space. For example, the vector field \( \mathbf{F}(x, y, z) \) is expressed as \( \langle F_1(x, y, z), F_2(x, y, z), F_3(x, y, z) \rangle \).

• The first component, \( F_1(x, y, z) \), describes how the vector field behaves along the x-axis.
• The second component, \( F_2(x, y, z) \), describes behavior along the y-axis.
• The third component, \( F_3(x, y, z) \), describes behavior along the z-axis.

These components may describe things like velocity in fluid flow or electrical forces in a physical space. Vector fields are ubiquitous in physics and engineering to model and analyze various phenomena.

Linearity of Operators

Linearity is a fundamental property in mathematics, where certain operations preserve the basic structure of expressions. When we say an operator is linear, it means the operator satisfies two main properties: additivity and scalar multiplication.
In the context of vector calculus, consider a vector field operator like the curl.
### AdditivityIf you apply the operator to the sum of two functions, it is the same as applying it to each function separately and then summing the results:
\( \operatorname{curl}(\mathbf{F} + \mathbf{G}) = \operatorname{curl}\mathbf{F} + \operatorname{curl}\mathbf{G} \).
### Scalar MultiplicationIf you multiply a function by a scalar before applying the operator, it is the same as applying the operator first and then multiplying by the scalar:
\( \operatorname{curl}(k\mathbf{F}) = k\operatorname{curl}\mathbf{F} \).
This property makes it easier to manipulate and compute with vector fields since complex expressions can be broken down into simpler parts.Applying these properties to the vector field operations greatly simplifies proofs and computations in engineering and physics.