Suppose that a vector field is given by $\vec{E} = \vec{A} e^{i \vec{k} \cdot \vec{r}}$, where $\vec{A}$ and $\vec{k}$ are constants and $\vec{r} = x \hat{x} + y \hat{y} + z \hat{z}$. Is it always true that the curl of $\vec{E}$ is equal to $i \vec{k} \times \vec{E}$? Or is this only true when $\vec{k}$ is perpendicular to $\vec{A}$?

Answered: Ãek-i, where Ã and k are constants and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Suppose that a vector field is given by $\vec{E} = \vec{A} e^{i \vec{k} \cdot \vec{r}}$, where $\vec{A}$ and $\vec{k}$ are constants and $\vec{r} = x \hat{x} + y \hat{y} + z \hat{z}$. Is it always true that the curl of $\vec{E}$ is equal to $i \vec{k} \times \vec{E}$? Or is this only true when $\vec{k}$ is perpendicular to $\vec{A}$?

Expert Solution

Step 1

A vector is a quantity that has both magnitude and direction. In general, vectors are represented as $\vec{A} = A_{1} \hat{i} + A_{2} \hat{j} + A_{3} \hat{k}$ , where $A_{1}, A_{2}, A_{3}$ are called the components of the vectors, and $\hat{i}, \hat{j}$ and $\hat{k}$ are the units vectors in the direction of x, y and z respectively. The dot product between two vectors $\vec{A}$ and $\vec{B}$ is defined as $\vec{A} \cdot \vec{B} = A_{1} B_{1} + A_{2} B_{2} + A_{3} B_{3}$

The partial derivative of a multivariable function with respect to a variable is the derivative of a function with respect to that particular variable, by treating all other variables as constants. For example, if $f (x, y, z) = x^{2} y^{2} z^{2}$ is a multivariable function, then the partial derivative of f with respect to x is $\frac{\partial f}{\partial x} = 2 x y^{2} z^{2}$ (since the derivative of $x^{2}$ with respect to x is $2 x$ and $y^{2} z^{2}$ is treated as constant while calculating the partial derivative).