Question

For the given vector field, find the divergence and curl of the field. a. \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}\) b. \(\mathbf{F}=\frac{y}{r} \mathbf{i}-\frac{x}{r} \mathbf{j}\), for \(r=\sqrt{x^{2}+y^{2}}\). c. \(\mathbf{F}=x^{2} y \mathbf{i}+z \mathbf{j}+x y z \mathbf{k} .\)

Short answer

The divergences are 2, 0, 0 respectively, and the curls are 0, 0, and \((1 - 2xy) \mathbf{k}\) respectively.

Step-by-step solution

Compute Divergence and Curl for Field F1

For the field \(F = x \mathbf{i} + y \mathbf{j}\), the divergence is computed as the dot product of the del operator and the vector field. Here, \(\nabla \cdot \mathbf{F} = \partial x/ \partial x +\partial y/ \partial y = 1 + 1 = 2\). For the curl, we need to calculate the cross product of the del operator and the field, leading to \(\nabla \times \mathbf{F} = i(\partial / \partial y - \partial / \partial z) - j(\partial / \partial x - \partial / \partial z) + k(\partial / \partial x - \partial / \partial y) \). Computing these differences gives \(\nabla \times \mathbf{F} = 0 \mathbf{i} - 0 \mathbf{j} + 0 \mathbf{k} = 0\)

Compute Divergence and Curl for Field F2

For the field \(F = \frac{y}{r} \mathbf{i} - \frac{x}{r} \mathbf{j}\) where \(r = \sqrt{x^2 + y^2}\), the divergence is \(\partial / \partial x (\frac{y}{r}) + \partial / \partial y (-\frac{x}{r})\). Substituting r and simplifying gives \(0\). The curl is computed similarly, leading to: \(\nabla \times \mathbf{F} = 0\)

Compute Divergence and Curl for Field F3

For the field \(F = x^2y\mathbf{i} + z\mathbf{j} + xyz\mathbf{k}\), the divergence is \(\partial / \partial x (x^2y) + \partial / \partial y (z) + \partial / \partial z (xyz) = 0\). The curl is \(\nabla \times \mathbf{F}\) = \(i(\frac{\partial}{\partial y}(xyz) - \frac{\partial}{\partial z}(z)) - j(\frac{\partial}{\partial x}(xyz) - \frac{\partial}{\partial z}(x^2y)) + k(\frac{\partial}{\partial x}(z) - \frac{\partial}{\partial y}(x^2y))\). Simplifying, we get \(\nabla \times \mathbf{F} = 0 \mathbf{i} - 0 \mathbf{j} + (1 - 2xy) \mathbf{k}\)

Key concepts

Divergence

Divergence is a key concept in vector calculus. It measures how much a vector field spreads out from a point. If you imagine a vector field representing the flow of a fluid, the divergence at a point tells you if the fluid is expanding or compressing at that point.

Mathematically, divergence uses the dot product to determine these changes. For instance, if you take the vector field \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}\), its divergence is calculated using \(abla \cdot \mathbf{F}\). You find this by taking the partial derivative of each component with respect to its variable: \(\partial/\partial x (x) + \partial/\partial y (y) = 1 + 1 = 2\). This indicates a consistent spreading out of the vector field in all directions.

For the second field \(F = \frac{y}{r} \mathbf{i} - \frac{x}{r} \mathbf{j}\), where \(r = \sqrt{x^2 + y^2}\), the divergence comes out as zero, meaning the field has no net source or sink at any point. It can be quite revealing about the underlying behavior of a vector field.

Curl

The curl of a vector field provides insight into its rotational characteristics. Consider it like analyzing how a field swirls around a point, like the whirl of water around a drain. The curl is defined as the cross product of the del operator with a vector field, usually symbolized as \(abla \times \mathbf{F}\).

For example, in the field \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}\), the curl is derived by computing the determinants related to each axis. Here, it simplifies to zero, indicating there is no rotation in the field; it is irrotational. Interestingly, for the second vector field \(F = \frac{y}{r} \mathbf{i} - \frac{x}{r} \mathbf{j}\), the same result is found, reinforcing its irrotational nature.

In contrast, when examining the third vector field \(F = x^2y\mathbf{i} + z\mathbf{j} + xyz\mathbf{k}\), the curl becomes non-zero only in the k-direction, resulting in \(abla \times \mathbf{F} = (1 - 2xy) \mathbf{k}\). It highlights a twist occurring in that direction, much like the last field where rotation isn't evenly distributed, showing complexity in vector rotation.

Vector Field Analysis

Vector field analysis is crucial for understanding and visualizing the flow and behavior of vectors in space. It combines the concepts of divergence and curl to provide a complete picture of field dynamics. Through analysis, one not only examines size and direction but also identifies behavior patterns, such as sources, sinks, or swirling motions.

When analyzing vector fields, it is important to pay attention to points of interest, such as where divergence is zero or non-zero, indicating whether the field is compressing, expanding, or neutral. Similarly, evaluating curl allows one to discern regions of rotational behavior versus more linear flow.

For practical applications, vector field analysis helps in areas such as fluid dynamics, electromagnetism, and atmospheric modeling. By accurately computing divergence and curl, you gain powerful insights into how these systems behave, predict changes, and even innovate solutions in engineering and science scenarios. It's fascinating how vector fields form the backbone of so many physical phenomena we seek to understand.