Finding Curl(F) of a vector-valued function

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In summary: So, the i-component of the curl of F is:\mathbf{i}\left ( \frac{\partial }{\partial y}-\frac{\partial }{\partial z}Qz\right )=\mathbf{\nabla}\times \mathbf{F}\left ( Ry-Qz\right )=\begin{vmatrix}& \mathbf{\alpha}& \mathbf{\beta}& \mathbf{\gamma}\\P(x,y,z)& Q(x,y,z)& R(x,y,z)\end{vmatrix}The determinant of this matrix
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Homework Statement



Determine the curl of the vector function below.

[tex]\boldsymbol{F}\left ( x,y,z \right )=3x^2\boldsymbol{i}+7e^xy\boldsymbol{j}[/tex]

Homework Equations



curl[itex]\mathbf{F}=\mathbf{\nabla}\times \mathbf{F}[/itex]
[tex]=\begin{vmatrix}
\mathbf{i}& \mathbf{j}& \mathbf{k}\\
\frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\
P(x,y,z)& Q(x,y,z)& R(x,y,z)
\end{vmatrix}[/tex]

The Attempt at a Solution



This problem is solved by my FE review as below. I understand how to solve for the determinant of a 3x3 matrix by rewriting the first two columns to the right of the matrix and obtaining three "+" terms and three "-" terms. I think my partials for x and y are off, perhaps.

[tex]\mathbf{i}\left ( \frac{\partial }{\partial y}0-\frac{\partial }{\partial z}7e^xy \right )-\mathbf{j}\left ( \frac{\partial }{\partial x}0-\frac{\partial }{\partial z}3x^2 \right )+\mathbf{k}\left ( \frac{\partial }{\partial x}7e^xy-\frac{\partial }{\partial y}3x^2 \right )[/tex]

[tex]=\mathbf{i}(0-0)-\mathbf{j}(0-0)-\mathbf{k}\left ( 7e^xy-0 \right )=7e^xy\mathbf{k}[/tex]

The expressions I calculated for [itex]\mathbf{i} and \mathbf{j}[/itex] match what the book has. However, my expression for [itex]\mathbf{k}[/itex] seems to be incorrect. Here is what I calculated for the values of the matrix:

[tex]\frac{\partial }{\partial x}=6x[/tex]
[tex]\frac{\partial }{\partial y}=7e^x[/tex]
[tex]\frac{\partial }{\partial z}=0[/tex]
[tex]P(x,y,z)=3x^2[/tex]
[tex]Q(x,y,z)=7e^xy[/tex]
[tex]R(x,y,z)=0[/tex]

So my expression for [itex]\mathbf{k}[/itex] was:

[tex]\left [(6x)7e^xy-\left ( 7e^x \right )3x^2 \right ]\mathbf{k}[/tex]

I think I went wrong with my calculation of [itex] \frac{\partial }{\partial x}[/itex].
 
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  • #2
It is unclear what exactly you have done.
My guess is that you found
$$\dfrac{\partial}{\partial x}3\, x^2=6\, x$$
when you should have found
$$\dfrac{\partial}{\partial x}7 \, e^x \, y=7 \, e^x \, y$$
also you are multiplying functions together for some reason

Look at the solution, of the six terms 5 are clearly zero so only the sixth remains.
 
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  • #3
This is why I love physicsforums. You answered me and helped me understand the problem within 20 minutes of my post.

You're right, I was multiplying the functions when I should not have been. Normally when working with matrices containing only constants, you *do* just multiply elements together. So that seems not to be the case when you have partial derivatives as elements.
 
  • #4
For the curl, it's helpful to think of the partial derivatives in the second row as operators to be applied to the functions P, Q, and R when the determinant is expanded. Expressing the curl in terms of a determinant is just a handy way to remember how to calculate its various components.

Taking the i-component of the curl, for example, expanding the determinant gives:

Ry - Qz, where the subscripts y and z indicate the variable w.r.t. which the partial derivative is taken.
 

1. What is the definition of curl in vector calculus?

Curl is a mathematical operation that measures the rotation or circulation of a vector field. It is represented by the symbol ∇ × F and is defined as the vector cross product of the gradient of a vector-valued function F.

2. How do you find the curl of a vector-valued function?

To find the curl of a vector-valued function, you first take the partial derivatives of each component of the function with respect to the x, y, and z coordinates. Then, you take the vector cross product of these derivatives to get the curl vector, which represents the rotation at any given point in the vector field.

3. What is the physical significance of curl?

Curl has many physical interpretations, but one of the most common is that it represents the flow of a fluid or the rotation of a rigid body. In electromagnetism, curl represents the circulation of an electric or magnetic field, which is important in understanding phenomena such as electromagnetic induction and Faraday's law.

4. Are there any applications of curl in real life?

Yes, there are many applications of curl in real life. Some examples include weather forecasting, fluid dynamics, and electromagnetic engineering. In weather forecasting, curl is used to model the circulation of air masses and predict the formation of storms. In fluid dynamics, curl is used to study the flow of fluids in pipes and around obstacles. In electromagnetic engineering, curl is used to analyze the behavior of electric and magnetic fields in different systems.

5. Can the curl of a vector-valued function be zero?

Yes, the curl of a vector-valued function can be zero. This occurs when the vector field is irrotational, meaning that there is no rotation at any point in the field. In other words, the field is conservative and can be described by a scalar potential function. This has important implications in physics, such as in the study of conservative forces and energy conservation.

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