Why do we use $Df$ rather than $f'$ for the derivative of a multivariable function?
2
$begingroup$
Is there any reason we use $Df$ for derivatives of multivariable functions but $f'$ derivatives of single variable functions despite having a definition that works for both:
$$Df(c) = f'(c) = L iff lim_{x to c} frac{f(x) - f(c) - L(x-c)}{||x-c||} = 0$$
derivatives notation math-history
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
asked Dec 8 '18 at 19:50
StefanStefan
2126
$endgroup$
add a comment |
2
$begingroup$
Is there any reason we use $Df$ for derivatives of multivariable functions but $f'$ derivatives of single variable functions despite having a definition that works for both:
$$Df(c) = f'(c) = L iff lim_{x to c} frac{f(x) - f(c) - L(x-c)}{||x-c||} = 0$$
derivatives notation math-history
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
asked Dec 8 '18 at 19:50
StefanStefan
2126
$endgroup$
2
$begingroup$
You could do this. I've seen people do $f'_x$ to refer to the $x$ partial though.
$endgroup$
– Alfred Yerger
Dec 8 '18 at 19:52
$begingroup$
I'm asking why the notation changed when multivariable calculus was created, rather than keeping the old $f'$ notation.
$endgroup$
– Stefan
Dec 8 '18 at 19:54
$begingroup$
The notation didn't necessarily change. $f'$ is Lagrange's notation and $Df$ is Euler's notation. Both of them had notations for multivariate functions, $f^{'}_{''}$ vs $D_{xy}$ for example. I'm not entirely sure if you can pin down a reason why Euler's is more popular for multiple variables, but I can guess that it's just do to clarity. And Lagrange's notation is clearer and more concise for a single variable.
$endgroup$
– Dando18
Dec 8 '18 at 20:01
$begingroup$
I think the main reason is that there's much more complexity in multivariate derivatives. You have total derivatives, partial derivatives, and the Jacobian. Spivak uses prime notation to refer to the Jacobian in "Calculus on Manifolds", so it's not entirely abandoned.
$endgroup$
– AlexanderJ93
Dec 8 '18 at 20:14
add a comment |
2
2
2
$begingroup$
Is there any reason we use $Df$ for derivatives of multivariable functions but $f'$ derivatives of single variable functions despite having a definition that works for both:
$$Df(c) = f'(c) = L iff lim_{x to c} frac{f(x) - f(c) - L(x-c)}{||x-c||} = 0$$
derivatives notation math-history
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
asked Dec 8 '18 at 19:50
StefanStefan
2126
$endgroup$
Is there any reason we use $Df$ for derivatives of multivariable functions but $f'$ derivatives of single variable functions despite having a definition that works for both:
$$Df(c) = f'(c) = L iff lim_{x to c} frac{f(x) - f(c) - L(x-c)}{||x-c||} = 0$$
derivatives notation math-history
derivatives notation math-history
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
asked Dec 8 '18 at 19:50
StefanStefan
2126
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
asked Dec 8 '18 at 19:50
StefanStefan
2126
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
edited Dec 8 '18 at 20:04
Rodrigo de Azevedo
13.1k41960
13.1k41960
asked Dec 8 '18 at 19:50
StefanStefan
2126
asked Dec 8 '18 at 19:50
StefanStefan
2126
asked Dec 8 '18 at 19:50
StefanStefan
2126
2126
2
$begingroup$
You could do this. I've seen people do $f'_x$ to refer to the $x$ partial though.
$endgroup$
– Alfred Yerger
Dec 8 '18 at 19:52
$begingroup$
I'm asking why the notation changed when multivariable calculus was created, rather than keeping the old $f'$ notation.
$endgroup$
– Stefan
Dec 8 '18 at 19:54
$begingroup$
The notation didn't necessarily change. $f'$ is Lagrange's notation and $Df$ is Euler's notation. Both of them had notations for multivariate functions, $f^{'}_{''}$ vs $D_{xy}$ for example. I'm not entirely sure if you can pin down a reason why Euler's is more popular for multiple variables, but I can guess that it's just do to clarity. And Lagrange's notation is clearer and more concise for a single variable.
$endgroup$
– Dando18
Dec 8 '18 at 20:01
$begingroup$
I think the main reason is that there's much more complexity in multivariate derivatives. You have total derivatives, partial derivatives, and the Jacobian. Spivak uses prime notation to refer to the Jacobian in "Calculus on Manifolds", so it's not entirely abandoned.
$endgroup$
– AlexanderJ93
Dec 8 '18 at 20:14
add a comment |
2
$begingroup$
You could do this. I've seen people do $f'_x$ to refer to the $x$ partial though.
$endgroup$
– Alfred Yerger
Dec 8 '18 at 19:52
$begingroup$
I'm asking why the notation changed when multivariable calculus was created, rather than keeping the old $f'$ notation.
$endgroup$
– Stefan
Dec 8 '18 at 19:54
$begingroup$
The notation didn't necessarily change. $f'$ is Lagrange's notation and $Df$ is Euler's notation. Both of them had notations for multivariate functions, $f^{'}_{''}$ vs $D_{xy}$ for example. I'm not entirely sure if you can pin down a reason why Euler's is more popular for multiple variables, but I can guess that it's just do to clarity. And Lagrange's notation is clearer and more concise for a single variable.
$endgroup$
– Dando18
Dec 8 '18 at 20:01
$begingroup$
I think the main reason is that there's much more complexity in multivariate derivatives. You have total derivatives, partial derivatives, and the Jacobian. Spivak uses prime notation to refer to the Jacobian in "Calculus on Manifolds", so it's not entirely abandoned.
$endgroup$
– AlexanderJ93
Dec 8 '18 at 20:14
2
2
$begingroup$
You could do this. I've seen people do $f'_x$ to refer to the $x$ partial though.
$endgroup$
– Alfred Yerger
Dec 8 '18 at 19:52
$begingroup$
You could do this. I've seen people do $f'_x$ to refer to the $x$ partial though.
$endgroup$
– Alfred Yerger
Dec 8 '18 at 19:52
$begingroup$
I'm asking why the notation changed when multivariable calculus was created, rather than keeping the old $f'$ notation.
$endgroup$
– Stefan
Dec 8 '18 at 19:54
$begingroup$
I'm asking why the notation changed when multivariable calculus was created, rather than keeping the old $f'$ notation.
$endgroup$
– Stefan
Dec 8 '18 at 19:54
$begingroup$
The notation didn't necessarily change. $f'$ is Lagrange's notation and $Df$ is Euler's notation. Both of them had notations for multivariate functions, $f^{'}_{''}$ vs $D_{xy}$ for example. I'm not entirely sure if you can pin down a reason why Euler's is more popular for multiple variables, but I can guess that it's just do to clarity. And Lagrange's notation is clearer and more concise for a single variable.
$endgroup$
– Dando18
Dec 8 '18 at 20:01
$begingroup$
The notation didn't necessarily change. $f'$ is Lagrange's notation and $Df$ is Euler's notation. Both of them had notations for multivariate functions, $f^{'}_{''}$ vs $D_{xy}$ for example. I'm not entirely sure if you can pin down a reason why Euler's is more popular for multiple variables, but I can guess that it's just do to clarity. And Lagrange's notation is clearer and more concise for a single variable.
$endgroup$
– Dando18
Dec 8 '18 at 20:01
$begingroup$
I think the main reason is that there's much more complexity in multivariate derivatives. You have total derivatives, partial derivatives, and the Jacobian. Spivak uses prime notation to refer to the Jacobian in "Calculus on Manifolds", so it's not entirely abandoned.
$endgroup$
– AlexanderJ93
Dec 8 '18 at 20:14
$begingroup$
I think the main reason is that there's much more complexity in multivariate derivatives. You have total derivatives, partial derivatives, and the Jacobian. Spivak uses prime notation to refer to the Jacobian in "Calculus on Manifolds", so it's not entirely abandoned.
$endgroup$
– AlexanderJ93
Dec 8 '18 at 20:14
add a comment |
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2
$begingroup$
You could do this. I've seen people do $f'_x$ to refer to the $x$ partial though.
$endgroup$
– Alfred Yerger
Dec 8 '18 at 19:52
$begingroup$
I'm asking why the notation changed when multivariable calculus was created, rather than keeping the old $f'$ notation.
$endgroup$
– Stefan
Dec 8 '18 at 19:54
$begingroup$
The notation didn't necessarily change. $f'$ is Lagrange's notation and $Df$ is Euler's notation. Both of them had notations for multivariate functions, $f^{'}_{''}$ vs $D_{xy}$ for example. I'm not entirely sure if you can pin down a reason why Euler's is more popular for multiple variables, but I can guess that it's just do to clarity. And Lagrange's notation is clearer and more concise for a single variable.
$endgroup$
– Dando18
Dec 8 '18 at 20:01
$begingroup$
I think the main reason is that there's much more complexity in multivariate derivatives. You have total derivatives, partial derivatives, and the Jacobian. Spivak uses prime notation to refer to the Jacobian in "Calculus on Manifolds", so it's not entirely abandoned.
$endgroup$
– AlexanderJ93
Dec 8 '18 at 20:14