Question

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

Short answer

In this exercise, we proved the four properties related to the divergence and curl of vector fields mentioned: a) The divergence of the sum of two vector fields \(\mathbf{F}\) and \(\mathbf{G}\) is equal to the sum of their individual divergences: \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\), b) The curl of the sum of two vector fields \(\mathbf{F}\) and \(\mathbf{G}\) is equal to the sum of their individual curls: \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\), c) The divergence of a scalar multiplied vector field \(c\mathbf{F}\) is equal to the scalar multiplied divergence of the vector field: \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\), d) The curl of a scalar multiplied vector field \(c\mathbf{F}\) is equal to the scalar multiplied curl of the vector field: \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\). We demonstrated each property by applying the definitions of divergence and curl using the del operator \(\nabla\), and exploiting the linearity of partial derivatives and the constant rule for derivatives.

Step-by-step solution

Proof of property (a)

To prove that \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\), we will start by calculating the divergence of \(\mathbf{F} + \mathbf{G}\). We have \(\nabla \cdot(\mathbf{F}+\mathbf{G}) = \nabla \cdot (\mathbf{F_1}+\mathbf{G_1},\mathbf{F_2}+\mathbf{G_2},\mathbf{F_3}+\mathbf{G_3})\), applying the definition of divergence, we get: \(\frac{\partial(\mathbf{F_1}+\mathbf{G_1})}{\partial{x}}+\frac{\partial(\mathbf{F_2}+\mathbf{G_2})}{\partial{y}}+\frac{\partial(\mathbf{F_3}+\mathbf{G_3})}{\partial{z}}\). Now we will use the linearity of partial derivatives: $ = \frac{\partial\mathbf{F_1}}{\partial{x}}+\frac{\partial\mathbf{G_1}}{\partial{x}}+\frac{\partial\mathbf{F_2}}{\partial{y}}+\frac{\partial\mathbf{G_2}}{\partial{y}}+\frac{\partial\mathbf{F_3}}{\partial{z}}+\frac{\partial\mathbf{G_3}}{\partial{z}}, $ which is equivalent to: $ (\frac{\partial\mathbf{F_1}}{\partial{x}}+\frac{\partial\mathbf{F_2}}{\partial{y}}+\frac{\partial\mathbf{F_3}}{\partial{z}})+(\frac{\partial\mathbf{G_1}}{\partial{x}}+\frac{\partial\mathbf{G_2}}{\partial{y}}+\frac{\partial\mathbf{G_3}}{\partial{z}}), $ Finally, we can write this as \(\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\).

Proof of property (b)

To prove that \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\), we will calculate the curl of \(\mathbf{F} + \mathbf{G}\) by taking the cross product of \(\nabla\) and \((\mathbf{F}+\mathbf{G})\): $ \nabla \times (\mathbf{F}+\mathbf{G}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}} \\ \mathbf{F_1}+\mathbf{G_1} & \mathbf{F_2}+\mathbf{G_2} & \mathbf{F_3}+\mathbf{G_3} \end{vmatrix}, $ Now we will expand this determinant: $ = \hat{i}\cdot(\frac{\partial(\mathbf{F_3}+\mathbf{G_3})}{\partial{y}}-\frac{\partial(\mathbf{F_2}+\mathbf{G_2})}{\partial{z}}) - \hat{j}\cdot(\frac{\partial(\mathbf{F_3}+\mathbf{G_3})}{\partial{x}}-\frac{\partial(\mathbf{F_1}+\mathbf{G_1})}{\partial{z}}) + \hat{k}\cdot(\frac{\partial(\mathbf{F_2}+\mathbf{G_2})}{\partial{x}}-\frac{\partial(\mathbf{F_1}+\mathbf{G_1})}{\partial{y}}). $ By applying linearity of partial derivatives, we can rewrite it as: \((\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\).

Proof of property (c)

To prove that \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\), we will calculate the divergence of the scaled vector field \(c\mathbf{F}\): $ \nabla \cdot(c \mathbf{F}) = \nabla \cdot (c\mathbf{F_1},c\mathbf{F_2},c\mathbf{F_3}), $ Applying the definition of divergence, we get: $ \frac{\partial(c\mathbf{F_1})}{\partial{x}}+\frac{\partial(c\mathbf{F_2})}{\partial{y}}+\frac{\partial(c\mathbf{F_3})}{\partial{z}}, $ Now we use the constant rule for derivatives: $ c(\frac{\partial\mathbf{F_1}}{\partial{x}}+ \frac{\partial\mathbf{F_2}}{\partial{y}}+\frac{\partial\mathbf{F_3}}{\partial{z}}), $ which is equal to \(c(\nabla \cdot \mathbf{F})\).

Proof of property (d)

To prove that \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\), we will calculate the curl of the scaled vector field \(c\mathbf{F}\) by taking the cross product of \(\nabla\) and \(c\mathbf{F}\): $ \nabla \times (c\mathbf{F}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}} \\ c\mathbf{F_1} & c\mathbf{F_2} & c\mathbf{F_3} \end{vmatrix}, $ Now we will expand this determinant and apply the constant rule for derivatives: $ = c\hat{i}\cdot(\frac{\partial\mathbf{F_3}}{\partial{y}}-\frac{\partial\mathbf{F_2}}{\partial{z}}) - c\hat{j}\cdot(\frac{\partial\mathbf{F_3}}{\partial{x}}-\frac{\partial\mathbf{F_1}}{\partial{z}}) + c\hat{k}\cdot(\frac{\partial\mathbf{F_2}}{\partial{x}}-\frac{\partial\mathbf{F_1}}{\partial{y}}). $ Finally, we can rewrite this as \(c(\nabla \times \mathbf{F})\).

Key concepts

Divergence

The divergence of a vector field is a measure of the field's tendency to originate from or converge into a point. In physics, it reflects how much a field spreads out from a point or converges into it, similar to the flow of a fluid. Mathematically, for a vector field \textbf{F} with components \textbf{F1}, \textbf{F2}, and \textbf{F3}, the divergence is expressed as:

\[\begin{equation}abla \bullet \textbf{F} = \frac{\partial \textbf{F1}}{\partial x} + \frac{\partial \textbf{F2}}{\partial y} + \frac{\partial \textbf{F3}}{\partial z}\end{equation}\]

This scalar quantity describes the 'source' or 'sink' strength at any given point in space. When we speak of the divergence of two vector fields added together, the linearity property ensures that the divergence of this sum can be calculated as the sum of their divergences independently. This property can be extended to the multiplication of a vector field by a scalar where the divergence of the scaled field is simply the scalar times the divergence of the original field.

Curl

Curl is a measure of the rotational or 'twisting' action of a vector field around a point. Analogous to a whirlpool in the water, curl gives us an idea of how and where the field 'rotates'. The vector field \textbf{F}'s curl at any point is represented as a vector and is calculated through the cross product of the del operator (∇) with the vector field \textbf{F}:

\[\begin{equation}abla \times \textbf{F}\end{equation}\]

The results of this calculation can be interpreted as the infinitesimal circulation intensity and direction around a point. If a vector field represents fluid flow, for instance, the curl at a point within the flow would indicate the axis of rotation and the magnitude of swirling at that point. Similar to divergence, the curl operator obeys distributive laws over vector addition and scaling, implying that the curl of the sum is the sum of the curls, and the curl of a scaled vector field can be represented as the scalar multiplier times the curl of the field.

Partial Derivatives

Partial derivatives are a fundamental concept used to assess how a function changes as one of its input variables is varied, holding all other input variables constant. It is like examining the slope of the landscape in only the north-south direction or the east-west direction, ignoring the overall terrain. When applied to vector fields in vector calculus, partial derivatives allow for the determination of both divergence and curl. In essence, while calculating the divergence or curl, we are taking partial derivatives of the vector field components with respect to their respective variables (x, y, and z). This leads to an understanding of how the field varies locally in three-dimensional space.

The proofs of properties involving divergence and curl heavily rely on the use of partial derivatives, leveraging their linearity which allows derivatives to be taken term-by-term when summing or scaling functions. Specifically, the gradient of a function or the partial derivatives of a vector field's components are what make up the components of divergence and curl.

Vector Fields

Vector fields are mathematical constructs that assign a vector to every point in space. A vector field might represent a variety of things depending on the context such as the force acting on objects due to gravity, or the speed and direction of the wind in the atmosphere. In the context of vector calculus, vector fields are central to the notions of divergence and curl.

Visualizing a vector field can involve picturing an arrow at each point in space, with the arrow's direction and length representing the direction and magnitude of the field. These vectors encapsulate both the magnitude and direction of some quantity at each point in space. Analyzing vector fields through divergence and curl provides insight into their behavior, allowing us to understand physical phenomena such as electromagnetic fields or fluid dynamics.

When we manipulate vector fields algebraically, as in the case of adding two fields or scaling a field, we're combining or adjusting the vectors associated with each point in space. This manipulation underlines the significance of vector fields in physics and engineering, helping us model and solve complex real-world problems.