Mixing
How do multiply the nabla operator by $f$?
$begingroup$
I have the function $f={x,-y,0}$, and each of the derivatives together result in ${1,-1,0}$. To calculate the divergence $nablacdot$ of $f$ I'd have to do the dot (scalar) product of partial derivatives with $f$, while for the curl $nablatimes$ I have to do the cross product. I
I know that the curl is ${0,0,2}$ but I do not really understand how this result comes up: the cross product of the partial derivatives with $f$ results in something like ${0,0,x+y}$.
Please bear in mind I'm very new to this topic and no one has taught me this, so it would help if someone could point out what has to be multiplied by what exactly. Thanks.
vectors vector-analysis vector-fields
edited Jan 7 at 13:32
amWhy
1
asked Jan 7 at 11:52
Display NameDisplay Name
91
$endgroup$
add a comment |
$begingroup$
I have the function $f={x,-y,0}$, and each of the derivatives together result in ${1,-1,0}$. To calculate the divergence $nablacdot$ of $f$ I'd have to do the dot (scalar) product of partial derivatives with $f$, while for the curl $nablatimes$ I have to do the cross product. I
I know that the curl is ${0,0,2}$ but I do not really understand how this result comes up: the cross product of the partial derivatives with $f$ results in something like ${0,0,x+y}$.
Please bear in mind I'm very new to this topic and no one has taught me this, so it would help if someone could point out what has to be multiplied by what exactly. Thanks.
vectors vector-analysis vector-fields
edited Jan 7 at 13:32
amWhy
1
asked Jan 7 at 11:52
Display NameDisplay Name
91
$endgroup$
$begingroup$
I noticed that I made a typo that changed the task into another one with the curl really being {0,0,0} like the comments say!
$endgroup$
– Display Name
Jan 14 at 20:56
add a comment |
$begingroup$
I have the function $f={x,-y,0}$, and each of the derivatives together result in ${1,-1,0}$. To calculate the divergence $nablacdot$ of $f$ I'd have to do the dot (scalar) product of partial derivatives with $f$, while for the curl $nablatimes$ I have to do the cross product. I
I know that the curl is ${0,0,2}$ but I do not really understand how this result comes up: the cross product of the partial derivatives with $f$ results in something like ${0,0,x+y}$.
Please bear in mind I'm very new to this topic and no one has taught me this, so it would help if someone could point out what has to be multiplied by what exactly. Thanks.
vectors vector-analysis vector-fields
edited Jan 7 at 13:32
amWhy
1
asked Jan 7 at 11:52
Display NameDisplay Name
91
$endgroup$
I have the function $f={x,-y,0}$, and each of the derivatives together result in ${1,-1,0}$. To calculate the divergence $nablacdot$ of $f$ I'd have to do the dot (scalar) product of partial derivatives with $f$, while for the curl $nablatimes$ I have to do the cross product. I
I know that the curl is ${0,0,2}$ but I do not really understand how this result comes up: the cross product of the partial derivatives with $f$ results in something like ${0,0,x+y}$.
Please bear in mind I'm very new to this topic and no one has taught me this, so it would help if someone could point out what has to be multiplied by what exactly. Thanks.
vectors vector-analysis vector-fields
vectors vector-analysis vector-fields
edited Jan 7 at 13:32
amWhy
1
asked Jan 7 at 11:52
Display NameDisplay Name
91
edited Jan 7 at 13:32
amWhy
1
asked Jan 7 at 11:52
Display NameDisplay Name
91
edited Jan 7 at 13:32
amWhy
1
edited Jan 7 at 13:32
amWhy
1
edited Jan 7 at 13:32
amWhy
1
1
asked Jan 7 at 11:52
Display NameDisplay Name
91
asked Jan 7 at 11:52
Display NameDisplay Name
91
asked Jan 7 at 11:52
Display NameDisplay Name
91
91
$begingroup$
I noticed that I made a typo that changed the task into another one with the curl really being {0,0,0} like the comments say!
$endgroup$
– Display Name
Jan 14 at 20:56
add a comment |
$begingroup$
I noticed that I made a typo that changed the task into another one with the curl really being {0,0,0} like the comments say!
$endgroup$
– Display Name
Jan 14 at 20:56
$begingroup$
I noticed that I made a typo that changed the task into another one with the curl really being {0,0,0} like the comments say!
$endgroup$
– Display Name
Jan 14 at 20:56
$begingroup$
I noticed that I made a typo that changed the task into another one with the curl really being {0,0,0} like the comments say!
$endgroup$
– Display Name
Jan 14 at 20:56
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
So what I understand is that you have trouble calculating the curl of a given vector field
$$
F_x = xqquad F_y=-y qquad F_z = 0
$$
In Wikipedia, for example, you can find the quite straightforward formula for the curl:
$$
nablatimes F = left( frac{partial F_z}{partial y} - frac{partial F_y}{partial z} right) hat{i}
+left( frac{partial F_x}{partial z} - frac{partial F_z}{partial x} right) hat{j}
+ left( frac{partial F_y}{partial x} - frac{partial F_x}{partial y} right) hat{k}
$$
As you see, there are no terms $frac{partial F_x}{partial x}$ or $frac{partial F_y}{partial y}$, so therefore the curl is zero.
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
$endgroup$
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
add a comment |
$begingroup$
Let's start from the definitions. We have a vector field (vector-valued function) $$bbox { begin{aligned}
mathbf{F}(x, y, z) &= bigr ( X(x, y, z) , Y(x, y, z) , Z(x, y, z) bigr ) \
; &= X(x,y,z) hat{mathbf{e}}_x + Y(x, y, z) hat{mathbf{e}}_y + Z(x, y, z) hat{mathbf{e}}_z \
end{aligned} }$$
where $X(x, y, z)$, $Y(x, y, z)$ and $Z(x, y, z)$ are scalar functions, and $hat{mathbf{e}}_x$, $hat{mathbf{e}}_y$, and $hat{mathbf{e}}_z$ are the standard unit vectors in the directions of the $x$, $y$, and $z$ coordinates, respectively.
Divergence is defined as
$$bbox {
nabla cdot mathbf{F}(x, y, z) = frac{partial X(x, y, z)}{partial x} + frac{partial Y(x, y, z)}{partial y} + frac{partial Z(x, y, z)}{partial z}
}$$
and curl is defined as
$$bbox { begin{aligned}
nabla times mathbf{F}(x, y, z) &= left [ begin{matrix}
hat{mathbf{e}}_x & hat{mathbf{e}}_y & hat{mathbf{e}}_z \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
X(x,y,z) & Y(x,y,z) & Z(x,y,z) \
end{matrix} right ] \
; &= hat{mathbf{e}}_x left (
frac{partial Z(x,y,z)}{partial y} - frac{partial Y(x,y,z)}{partial z}
right ) \
; &, + hat{mathbf{e}}_y left (
frac{partial X(x,y,z)}{partial z} - frac{partial Z(x,y,z)}{partial x}
right ) \
; &, + hat{mathbf{e}}_z left (
frac{partial Y(x,y,z)}{partial x} - frac{partial X(x,y,z)}{partial y}
right ) \
end{aligned} }$$
These are the same things if you wrote $nabla = left ( frac{partial}{partial x} , frac{partial}{partial y}, frac{partial}{partial z} right )$ and did the dot and cross products, respectively.
In OP's case, $X(x,y,z) = x$, $Y(x,y,z) = -y$, and $Z(x,y,z) = 0$.
For divergence, that gives us
$$bbox{ begin{aligned}
nabla cdot mathbf{F}(x, y, z) &= frac{d x}{d x} + frac{d(-y)}{d y} + 0 \
; &= 1 - 1 \
; &= 0 \
end{aligned} }$$
For curl, we have
$$bbox{ begin{aligned}
nabla times mathbf{F}(x, y, z) &= hat{mathbf{e}}_x bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_y bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_z bigr ( 0 - 0 bigr ) \
; & = bigr ( 0, 0, 0 bigr ) \
end{aligned} }$$
Note that Maple agrees,
> with(VectorCalculus):
> f := VectorField(<x,-y,0>, 'cartesian'[x, y, z]):
> Divergence(f);
0
> Curl(f);
_ _ _
(0)e + (0)e + (0)e
x y z
and so does SageMath:
sage: var('x y z')
sage: f = vector([x, -y, 0])
sage: f.div([x, y, z])
0
sage: f.curl([x, y, z])
(0, 0, 0)
They are rather nice tools to use to verify your calculations.
Note that I myself am not a mathematician, but use math as a tool constantly. Even though I use Maple and SageMath almost exclusively to do the hard work, it only works if I know the rules and operations and methods. Maple and SageMath and other tools are definitely not a replacement for learning the math; but when you do have a grasp on the math, they do save a lot of time, and help you avoid most errors. (They do not help you avoid logical errors, like applying the wrong tool or algorithm or solution method to the wrong problem; which is why there is no alternative to learning math.)
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
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add a comment |
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2 Answers
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2 Answers
2
active
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$begingroup$
So what I understand is that you have trouble calculating the curl of a given vector field
$$
F_x = xqquad F_y=-y qquad F_z = 0
$$
In Wikipedia, for example, you can find the quite straightforward formula for the curl:
$$
nablatimes F = left( frac{partial F_z}{partial y} - frac{partial F_y}{partial z} right) hat{i}
+left( frac{partial F_x}{partial z} - frac{partial F_z}{partial x} right) hat{j}
+ left( frac{partial F_y}{partial x} - frac{partial F_x}{partial y} right) hat{k}
$$
As you see, there are no terms $frac{partial F_x}{partial x}$ or $frac{partial F_y}{partial y}$, so therefore the curl is zero.
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
$endgroup$
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
add a comment |
$begingroup$
So what I understand is that you have trouble calculating the curl of a given vector field
$$
F_x = xqquad F_y=-y qquad F_z = 0
$$
In Wikipedia, for example, you can find the quite straightforward formula for the curl:
$$
nablatimes F = left( frac{partial F_z}{partial y} - frac{partial F_y}{partial z} right) hat{i}
+left( frac{partial F_x}{partial z} - frac{partial F_z}{partial x} right) hat{j}
+ left( frac{partial F_y}{partial x} - frac{partial F_x}{partial y} right) hat{k}
$$
As you see, there are no terms $frac{partial F_x}{partial x}$ or $frac{partial F_y}{partial y}$, so therefore the curl is zero.
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
$endgroup$
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
add a comment |
$begingroup$
So what I understand is that you have trouble calculating the curl of a given vector field
$$
F_x = xqquad F_y=-y qquad F_z = 0
$$
In Wikipedia, for example, you can find the quite straightforward formula for the curl:
$$
nablatimes F = left( frac{partial F_z}{partial y} - frac{partial F_y}{partial z} right) hat{i}
+left( frac{partial F_x}{partial z} - frac{partial F_z}{partial x} right) hat{j}
+ left( frac{partial F_y}{partial x} - frac{partial F_x}{partial y} right) hat{k}
$$
As you see, there are no terms $frac{partial F_x}{partial x}$ or $frac{partial F_y}{partial y}$, so therefore the curl is zero.
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
$endgroup$
So what I understand is that you have trouble calculating the curl of a given vector field
$$
F_x = xqquad F_y=-y qquad F_z = 0
$$
In Wikipedia, for example, you can find the quite straightforward formula for the curl:
$$
nablatimes F = left( frac{partial F_z}{partial y} - frac{partial F_y}{partial z} right) hat{i}
+left( frac{partial F_x}{partial z} - frac{partial F_z}{partial x} right) hat{j}
+ left( frac{partial F_y}{partial x} - frac{partial F_x}{partial y} right) hat{k}
$$
As you see, there are no terms $frac{partial F_x}{partial x}$ or $frac{partial F_y}{partial y}$, so therefore the curl is zero.
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
edited Jan 7 at 12:21
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
answered Jan 7 at 12:08
Matti P.Matti P.
1,881413
1,881413
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
add a comment |
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Hello and thanks for your answer, Maybe I didn't word it so clearly but I do know the Formula and also the curl is a Vector {0,0,2} . I was asking for a calculation of that Vector cross products in detail not with the Formula Units but the actual digits that apply for this example.
$endgroup$
– Display Name
Jan 7 at 17:45
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
$begingroup$
Is the fector field $F$ correct? Because I'm quite convinced that the curl is zero (more specifically, it's $left{ 0, 0, 0 right}$ and independent of the values of $x, y, z$). But you say that you "know that the curl is $left{ 0, 0, 2 right}$" - how do you get this result? And how did you get the result $left{ 0, 0, x+y right}$ ?
$endgroup$
– Matti P.
Jan 8 at 6:16
add a comment |
$begingroup$
Let's start from the definitions. We have a vector field (vector-valued function) $$bbox { begin{aligned}
mathbf{F}(x, y, z) &= bigr ( X(x, y, z) , Y(x, y, z) , Z(x, y, z) bigr ) \
; &= X(x,y,z) hat{mathbf{e}}_x + Y(x, y, z) hat{mathbf{e}}_y + Z(x, y, z) hat{mathbf{e}}_z \
end{aligned} }$$
where $X(x, y, z)$, $Y(x, y, z)$ and $Z(x, y, z)$ are scalar functions, and $hat{mathbf{e}}_x$, $hat{mathbf{e}}_y$, and $hat{mathbf{e}}_z$ are the standard unit vectors in the directions of the $x$, $y$, and $z$ coordinates, respectively.
Divergence is defined as
$$bbox {
nabla cdot mathbf{F}(x, y, z) = frac{partial X(x, y, z)}{partial x} + frac{partial Y(x, y, z)}{partial y} + frac{partial Z(x, y, z)}{partial z}
}$$
and curl is defined as
$$bbox { begin{aligned}
nabla times mathbf{F}(x, y, z) &= left [ begin{matrix}
hat{mathbf{e}}_x & hat{mathbf{e}}_y & hat{mathbf{e}}_z \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
X(x,y,z) & Y(x,y,z) & Z(x,y,z) \
end{matrix} right ] \
; &= hat{mathbf{e}}_x left (
frac{partial Z(x,y,z)}{partial y} - frac{partial Y(x,y,z)}{partial z}
right ) \
; &, + hat{mathbf{e}}_y left (
frac{partial X(x,y,z)}{partial z} - frac{partial Z(x,y,z)}{partial x}
right ) \
; &, + hat{mathbf{e}}_z left (
frac{partial Y(x,y,z)}{partial x} - frac{partial X(x,y,z)}{partial y}
right ) \
end{aligned} }$$
These are the same things if you wrote $nabla = left ( frac{partial}{partial x} , frac{partial}{partial y}, frac{partial}{partial z} right )$ and did the dot and cross products, respectively.
In OP's case, $X(x,y,z) = x$, $Y(x,y,z) = -y$, and $Z(x,y,z) = 0$.
For divergence, that gives us
$$bbox{ begin{aligned}
nabla cdot mathbf{F}(x, y, z) &= frac{d x}{d x} + frac{d(-y)}{d y} + 0 \
; &= 1 - 1 \
; &= 0 \
end{aligned} }$$
For curl, we have
$$bbox{ begin{aligned}
nabla times mathbf{F}(x, y, z) &= hat{mathbf{e}}_x bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_y bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_z bigr ( 0 - 0 bigr ) \
; & = bigr ( 0, 0, 0 bigr ) \
end{aligned} }$$
Note that Maple agrees,
> with(VectorCalculus):
> f := VectorField(<x,-y,0>, 'cartesian'[x, y, z]):
> Divergence(f);
0
> Curl(f);
_ _ _
(0)e + (0)e + (0)e
x y z
and so does SageMath:
sage: var('x y z')
sage: f = vector([x, -y, 0])
sage: f.div([x, y, z])
0
sage: f.curl([x, y, z])
(0, 0, 0)
They are rather nice tools to use to verify your calculations.
Note that I myself am not a mathematician, but use math as a tool constantly. Even though I use Maple and SageMath almost exclusively to do the hard work, it only works if I know the rules and operations and methods. Maple and SageMath and other tools are definitely not a replacement for learning the math; but when you do have a grasp on the math, they do save a lot of time, and help you avoid most errors. (They do not help you avoid logical errors, like applying the wrong tool or algorithm or solution method to the wrong problem; which is why there is no alternative to learning math.)
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
$endgroup$
add a comment |
$begingroup$
Let's start from the definitions. We have a vector field (vector-valued function) $$bbox { begin{aligned}
mathbf{F}(x, y, z) &= bigr ( X(x, y, z) , Y(x, y, z) , Z(x, y, z) bigr ) \
; &= X(x,y,z) hat{mathbf{e}}_x + Y(x, y, z) hat{mathbf{e}}_y + Z(x, y, z) hat{mathbf{e}}_z \
end{aligned} }$$
where $X(x, y, z)$, $Y(x, y, z)$ and $Z(x, y, z)$ are scalar functions, and $hat{mathbf{e}}_x$, $hat{mathbf{e}}_y$, and $hat{mathbf{e}}_z$ are the standard unit vectors in the directions of the $x$, $y$, and $z$ coordinates, respectively.
Divergence is defined as
$$bbox {
nabla cdot mathbf{F}(x, y, z) = frac{partial X(x, y, z)}{partial x} + frac{partial Y(x, y, z)}{partial y} + frac{partial Z(x, y, z)}{partial z}
}$$
and curl is defined as
$$bbox { begin{aligned}
nabla times mathbf{F}(x, y, z) &= left [ begin{matrix}
hat{mathbf{e}}_x & hat{mathbf{e}}_y & hat{mathbf{e}}_z \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
X(x,y,z) & Y(x,y,z) & Z(x,y,z) \
end{matrix} right ] \
; &= hat{mathbf{e}}_x left (
frac{partial Z(x,y,z)}{partial y} - frac{partial Y(x,y,z)}{partial z}
right ) \
; &, + hat{mathbf{e}}_y left (
frac{partial X(x,y,z)}{partial z} - frac{partial Z(x,y,z)}{partial x}
right ) \
; &, + hat{mathbf{e}}_z left (
frac{partial Y(x,y,z)}{partial x} - frac{partial X(x,y,z)}{partial y}
right ) \
end{aligned} }$$
These are the same things if you wrote $nabla = left ( frac{partial}{partial x} , frac{partial}{partial y}, frac{partial}{partial z} right )$ and did the dot and cross products, respectively.
In OP's case, $X(x,y,z) = x$, $Y(x,y,z) = -y$, and $Z(x,y,z) = 0$.
For divergence, that gives us
$$bbox{ begin{aligned}
nabla cdot mathbf{F}(x, y, z) &= frac{d x}{d x} + frac{d(-y)}{d y} + 0 \
; &= 1 - 1 \
; &= 0 \
end{aligned} }$$
For curl, we have
$$bbox{ begin{aligned}
nabla times mathbf{F}(x, y, z) &= hat{mathbf{e}}_x bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_y bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_z bigr ( 0 - 0 bigr ) \
; & = bigr ( 0, 0, 0 bigr ) \
end{aligned} }$$
Note that Maple agrees,
> with(VectorCalculus):
> f := VectorField(<x,-y,0>, 'cartesian'[x, y, z]):
> Divergence(f);
0
> Curl(f);
_ _ _
(0)e + (0)e + (0)e
x y z
and so does SageMath:
sage: var('x y z')
sage: f = vector([x, -y, 0])
sage: f.div([x, y, z])
0
sage: f.curl([x, y, z])
(0, 0, 0)
They are rather nice tools to use to verify your calculations.
Note that I myself am not a mathematician, but use math as a tool constantly. Even though I use Maple and SageMath almost exclusively to do the hard work, it only works if I know the rules and operations and methods. Maple and SageMath and other tools are definitely not a replacement for learning the math; but when you do have a grasp on the math, they do save a lot of time, and help you avoid most errors. (They do not help you avoid logical errors, like applying the wrong tool or algorithm or solution method to the wrong problem; which is why there is no alternative to learning math.)
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
$endgroup$
add a comment |
$begingroup$
Let's start from the definitions. We have a vector field (vector-valued function) $$bbox { begin{aligned}
mathbf{F}(x, y, z) &= bigr ( X(x, y, z) , Y(x, y, z) , Z(x, y, z) bigr ) \
; &= X(x,y,z) hat{mathbf{e}}_x + Y(x, y, z) hat{mathbf{e}}_y + Z(x, y, z) hat{mathbf{e}}_z \
end{aligned} }$$
where $X(x, y, z)$, $Y(x, y, z)$ and $Z(x, y, z)$ are scalar functions, and $hat{mathbf{e}}_x$, $hat{mathbf{e}}_y$, and $hat{mathbf{e}}_z$ are the standard unit vectors in the directions of the $x$, $y$, and $z$ coordinates, respectively.
Divergence is defined as
$$bbox {
nabla cdot mathbf{F}(x, y, z) = frac{partial X(x, y, z)}{partial x} + frac{partial Y(x, y, z)}{partial y} + frac{partial Z(x, y, z)}{partial z}
}$$
and curl is defined as
$$bbox { begin{aligned}
nabla times mathbf{F}(x, y, z) &= left [ begin{matrix}
hat{mathbf{e}}_x & hat{mathbf{e}}_y & hat{mathbf{e}}_z \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
X(x,y,z) & Y(x,y,z) & Z(x,y,z) \
end{matrix} right ] \
; &= hat{mathbf{e}}_x left (
frac{partial Z(x,y,z)}{partial y} - frac{partial Y(x,y,z)}{partial z}
right ) \
; &, + hat{mathbf{e}}_y left (
frac{partial X(x,y,z)}{partial z} - frac{partial Z(x,y,z)}{partial x}
right ) \
; &, + hat{mathbf{e}}_z left (
frac{partial Y(x,y,z)}{partial x} - frac{partial X(x,y,z)}{partial y}
right ) \
end{aligned} }$$
These are the same things if you wrote $nabla = left ( frac{partial}{partial x} , frac{partial}{partial y}, frac{partial}{partial z} right )$ and did the dot and cross products, respectively.
In OP's case, $X(x,y,z) = x$, $Y(x,y,z) = -y$, and $Z(x,y,z) = 0$.
For divergence, that gives us
$$bbox{ begin{aligned}
nabla cdot mathbf{F}(x, y, z) &= frac{d x}{d x} + frac{d(-y)}{d y} + 0 \
; &= 1 - 1 \
; &= 0 \
end{aligned} }$$
For curl, we have
$$bbox{ begin{aligned}
nabla times mathbf{F}(x, y, z) &= hat{mathbf{e}}_x bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_y bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_z bigr ( 0 - 0 bigr ) \
; & = bigr ( 0, 0, 0 bigr ) \
end{aligned} }$$
Note that Maple agrees,
> with(VectorCalculus):
> f := VectorField(<x,-y,0>, 'cartesian'[x, y, z]):
> Divergence(f);
0
> Curl(f);
_ _ _
(0)e + (0)e + (0)e
x y z
and so does SageMath:
sage: var('x y z')
sage: f = vector([x, -y, 0])
sage: f.div([x, y, z])
0
sage: f.curl([x, y, z])
(0, 0, 0)
They are rather nice tools to use to verify your calculations.
Note that I myself am not a mathematician, but use math as a tool constantly. Even though I use Maple and SageMath almost exclusively to do the hard work, it only works if I know the rules and operations and methods. Maple and SageMath and other tools are definitely not a replacement for learning the math; but when you do have a grasp on the math, they do save a lot of time, and help you avoid most errors. (They do not help you avoid logical errors, like applying the wrong tool or algorithm or solution method to the wrong problem; which is why there is no alternative to learning math.)
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
$endgroup$
Let's start from the definitions. We have a vector field (vector-valued function) $$bbox { begin{aligned}
mathbf{F}(x, y, z) &= bigr ( X(x, y, z) , Y(x, y, z) , Z(x, y, z) bigr ) \
; &= X(x,y,z) hat{mathbf{e}}_x + Y(x, y, z) hat{mathbf{e}}_y + Z(x, y, z) hat{mathbf{e}}_z \
end{aligned} }$$
where $X(x, y, z)$, $Y(x, y, z)$ and $Z(x, y, z)$ are scalar functions, and $hat{mathbf{e}}_x$, $hat{mathbf{e}}_y$, and $hat{mathbf{e}}_z$ are the standard unit vectors in the directions of the $x$, $y$, and $z$ coordinates, respectively.
Divergence is defined as
$$bbox {
nabla cdot mathbf{F}(x, y, z) = frac{partial X(x, y, z)}{partial x} + frac{partial Y(x, y, z)}{partial y} + frac{partial Z(x, y, z)}{partial z}
}$$
and curl is defined as
$$bbox { begin{aligned}
nabla times mathbf{F}(x, y, z) &= left [ begin{matrix}
hat{mathbf{e}}_x & hat{mathbf{e}}_y & hat{mathbf{e}}_z \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
X(x,y,z) & Y(x,y,z) & Z(x,y,z) \
end{matrix} right ] \
; &= hat{mathbf{e}}_x left (
frac{partial Z(x,y,z)}{partial y} - frac{partial Y(x,y,z)}{partial z}
right ) \
; &, + hat{mathbf{e}}_y left (
frac{partial X(x,y,z)}{partial z} - frac{partial Z(x,y,z)}{partial x}
right ) \
; &, + hat{mathbf{e}}_z left (
frac{partial Y(x,y,z)}{partial x} - frac{partial X(x,y,z)}{partial y}
right ) \
end{aligned} }$$
These are the same things if you wrote $nabla = left ( frac{partial}{partial x} , frac{partial}{partial y}, frac{partial}{partial z} right )$ and did the dot and cross products, respectively.
In OP's case, $X(x,y,z) = x$, $Y(x,y,z) = -y$, and $Z(x,y,z) = 0$.
For divergence, that gives us
$$bbox{ begin{aligned}
nabla cdot mathbf{F}(x, y, z) &= frac{d x}{d x} + frac{d(-y)}{d y} + 0 \
; &= 1 - 1 \
; &= 0 \
end{aligned} }$$
For curl, we have
$$bbox{ begin{aligned}
nabla times mathbf{F}(x, y, z) &= hat{mathbf{e}}_x bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_y bigr ( 0 - 0 bigr ) \
; & + , hat{mathbf{e}}_z bigr ( 0 - 0 bigr ) \
; & = bigr ( 0, 0, 0 bigr ) \
end{aligned} }$$
Note that Maple agrees,
> with(VectorCalculus):
> f := VectorField(<x,-y,0>, 'cartesian'[x, y, z]):
> Divergence(f);
0
> Curl(f);
_ _ _
(0)e + (0)e + (0)e
x y z
and so does SageMath:
sage: var('x y z')
sage: f = vector([x, -y, 0])
sage: f.div([x, y, z])
0
sage: f.curl([x, y, z])
(0, 0, 0)
They are rather nice tools to use to verify your calculations.
Note that I myself am not a mathematician, but use math as a tool constantly. Even though I use Maple and SageMath almost exclusively to do the hard work, it only works if I know the rules and operations and methods. Maple and SageMath and other tools are definitely not a replacement for learning the math; but when you do have a grasp on the math, they do save a lot of time, and help you avoid most errors. (They do not help you avoid logical errors, like applying the wrong tool or algorithm or solution method to the wrong problem; which is why there is no alternative to learning math.)
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
answered Jan 7 at 19:16
Nominal AnimalNominal Animal
7,0232517
7,0232517
add a comment |
add a comment |
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$begingroup$
I noticed that I made a typo that changed the task into another one with the curl really being {0,0,0} like the comments say!
$endgroup$
– Display Name
Jan 14 at 20:56