Problem 32 - 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. $$ \operatorname{curl}(k \mathbf{F})=k \operatorname{curl} \mathbf{F} $$

Step-by-Step Solution

Answer

The curl of a scaled vector is the scale times the curl: \( \operatorname{curl}(k \mathbf{F}) = k \operatorname{curl}(\mathbf{F}) \).

Step 1: Understand the Curl Operator

The curl of a vector field \( \mathbf{F} = P\hat{i} + Q\hat{j} + R\hat{k} \) is a vector field given by:\[ \operatorname{curl}(\mathbf{F}) = abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \hat{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \hat{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \hat{k} \]

Step 2: Apply the Curl to the Scaled Function

The expression \( k \mathbf{F} = k(P\hat{i} + Q\hat{j} + R\hat{k}) \) is \( kP\hat{i} + kQ\hat{j} + kR\hat{k} \). The curl of this vector field is:\[ \operatorname{curl}(k \mathbf{F}) = abla \times (k \mathbf{F}) = \left(\frac{\partial (kR)}{\partial y} - \frac{\partial (kQ)}{\partial z}\right) \hat{i} + \left(\frac{\partial (kP)}{\partial z} - \frac{\partial (kR)}{\partial x}\right) \hat{j} + \left(\frac{\partial (kQ)}{\partial x} - \frac{\partial (kP)}{\partial y}\right) \hat{k} \]

Step 3: Differentiate and Factor the Constant

By the linearity of differentiation, we can pull the constant \( k \) out of each partial derivative:\[ \operatorname{curl}(k \mathbf{F}) = k \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \hat{i} + k \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \hat{j} + k \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \hat{k} \]

Step 4: Simplify the Expression

Notice that each component is just \( k \) times the corresponding component of \( \operatorname{curl}(\mathbf{F}) \). Therefore, we have:\[ \operatorname{curl}(k \mathbf{F}) = k \operatorname{curl}(\mathbf{F}) \]

Step 5: Conclude the Proof

The identity \( \operatorname{curl}(k \mathbf{F}) = k \operatorname{curl}(\mathbf{F}) \) is proven. The curl of a scaled vector field is equivalent to scaling the curl by the same constant. This applies due to the linearity of partial derivatives.

Key Concepts

Linearity of Differentiation

Linearity of differentiation is a fundamental property in calculus, ensuring that derivatives of functions respect both addition and scalar multiplication. This is crucial for simplifying complex calculus problems. When we have differentiable functions, say \( f(x) \) and \( g(x) \), and a constant \( c \), the linearity property ensures:

• The derivative of a sum is the sum of the derivatives: \((f + g)' = f' + g'\).
• The derivative of a constant times a function is the constant times the derivative of the function: \((cf)' = c \cdot f'\).

In the context of vector calculus, this linearity extends to partial derivatives and differential operators like the curl. This means you can factor constants out of the curl operator, just as you would with simpler derivatives. This principle was key in proving our given identity where the constant multiplier \( k \) was factored outside each partial derivative. Hence, giving us the expression \( \operatorname{curl}(k \mathbf{F}) = k \operatorname{curl}(\mathbf{F}) \). This is a direct consequence of the linear behavior of differentiation.

Partial Derivatives

Partial derivatives are a way to explore the behavior of multivariable functions. When a function depends on several variables, the partial derivative with respect to one of those variables indicates how the function changes if that variable changes, while all others are held constant. For example, if we have a function \( f(x, y, z) \), then the partial derivative with respect to \( x \) is notated as \( \frac{\partial f}{\partial x} \). This focuses purely on changes along the \( x \)-axis.
In the context of vector fields, like \( \mathbf{F} = P\hat{i} + Q\hat{j} + R\hat{k} \), partial derivatives help us evaluate how each component of the field varies in different directions. The curl of a vector field utilizes these derivatives to measure the field's rotation by computing partial derivatives of each component against the orthogonal coordinates. This concept aids in analyzing fluid dynamics, electromagnetism, and more.

Proof in Vector Calculus

Proofs in vector calculus ensure that the mathematical statements we use are verifiable and reliable. These proofs often involve demonstrating that a particular identity holds under certain assumptions, like continuity and differentiability. For example, in the stated exercise, we aim to prove \( \operatorname{curl}(k \mathbf{F}) = k \operatorname{curl}(\mathbf{F}) \). Steps are methodical:

• Start by applying the vector calculus operation—in this case, curl—to the scaled vector field.
• Notice how the operation interacts with scalar multiplication, invoking the linearity of differentiation.
• Extract the scalar constant from the operation, simplifying the expression.
• Compare with the expected result and conclude the proof.

This rigorous process ensures the findings align with established mathematical rules, reinforcing the reliability of the calculus operations performed.