Partial Derivative with Respect to Multiple Variables
If we take a multivariable function such as $w=f(x,y,z)=x^2+y^2+z^2$, I understand that we can take its partial derivative with respect to any one of its arguments, while the others stay unchanged. In this case, we can take the partial derivative with respect to $z$ as $frac{partial}{partial z}f(x,y,z)$. I also understand that we can take its total derivative which is with respect to all of its arguments, which can be expressed as $frac{dt}{dw}f(x,y,z)$.
My question: how do we represent the derivative with respect to some but not all of a multivariable function's variables? How is this derivative classified? I am tempted to think it is partial but I am not sure since all of the definitions I have seen give it with respect to a single variable. Is it a "partial" total derivative?
multivariable-calculus partial-derivative
asked yesterday
Gnumbertester
505
add a comment |
If we take a multivariable function such as $w=f(x,y,z)=x^2+y^2+z^2$, I understand that we can take its partial derivative with respect to any one of its arguments, while the others stay unchanged. In this case, we can take the partial derivative with respect to $z$ as $frac{partial}{partial z}f(x,y,z)$. I also understand that we can take its total derivative which is with respect to all of its arguments, which can be expressed as $frac{dt}{dw}f(x,y,z)$.
My question: how do we represent the derivative with respect to some but not all of a multivariable function's variables? How is this derivative classified? I am tempted to think it is partial but I am not sure since all of the definitions I have seen give it with respect to a single variable. Is it a "partial" total derivative?
multivariable-calculus partial-derivative
asked yesterday
Gnumbertester
505
1
Note: a total derivative would be something like $$frac{dw}{dt}=frac{partial w}{partial x}frac{dx}{dt}+frac{partial w}{partial y}frac{dy}{dt}+frac{partial w}{partial z}frac{dz}{dt}$$
– John Doe
yesterday
See mixed partial derivatives
– WaveX
yesterday
Your notation with the "total derivative" makes absolutely no sense. I assume you mean $frac d{dt} f(x(t),y(t),z(t))$ or, often, $frac d{dt} f(t,x(t),y(t),z(t))$.
– Ted Shifrin
yesterday
add a comment |
If we take a multivariable function such as $w=f(x,y,z)=x^2+y^2+z^2$, I understand that we can take its partial derivative with respect to any one of its arguments, while the others stay unchanged. In this case, we can take the partial derivative with respect to $z$ as $frac{partial}{partial z}f(x,y,z)$. I also understand that we can take its total derivative which is with respect to all of its arguments, which can be expressed as $frac{dt}{dw}f(x,y,z)$.
My question: how do we represent the derivative with respect to some but not all of a multivariable function's variables? How is this derivative classified? I am tempted to think it is partial but I am not sure since all of the definitions I have seen give it with respect to a single variable. Is it a "partial" total derivative?
multivariable-calculus partial-derivative
asked yesterday
Gnumbertester
505
If we take a multivariable function such as $w=f(x,y,z)=x^2+y^2+z^2$, I understand that we can take its partial derivative with respect to any one of its arguments, while the others stay unchanged. In this case, we can take the partial derivative with respect to $z$ as $frac{partial}{partial z}f(x,y,z)$. I also understand that we can take its total derivative which is with respect to all of its arguments, which can be expressed as $frac{dt}{dw}f(x,y,z)$.
My question: how do we represent the derivative with respect to some but not all of a multivariable function's variables? How is this derivative classified? I am tempted to think it is partial but I am not sure since all of the definitions I have seen give it with respect to a single variable. Is it a "partial" total derivative?
multivariable-calculus partial-derivative
multivariable-calculus partial-derivative
asked yesterday
Gnumbertester
505
asked yesterday
Gnumbertester
505
edited yesterday
asked yesterday
Gnumbertester
505
asked yesterday
Gnumbertester
505
asked yesterday
Gnumbertester
505
505
1
Note: a total derivative would be something like $$frac{dw}{dt}=frac{partial w}{partial x}frac{dx}{dt}+frac{partial w}{partial y}frac{dy}{dt}+frac{partial w}{partial z}frac{dz}{dt}$$
– John Doe
yesterday
See mixed partial derivatives
– WaveX
yesterday
Your notation with the "total derivative" makes absolutely no sense. I assume you mean $frac d{dt} f(x(t),y(t),z(t))$ or, often, $frac d{dt} f(t,x(t),y(t),z(t))$.
– Ted Shifrin
yesterday
add a comment |
1
Note: a total derivative would be something like $$frac{dw}{dt}=frac{partial w}{partial x}frac{dx}{dt}+frac{partial w}{partial y}frac{dy}{dt}+frac{partial w}{partial z}frac{dz}{dt}$$
– John Doe
yesterday
See mixed partial derivatives
– WaveX
yesterday
Your notation with the "total derivative" makes absolutely no sense. I assume you mean $frac d{dt} f(x(t),y(t),z(t))$ or, often, $frac d{dt} f(t,x(t),y(t),z(t))$.
– Ted Shifrin
yesterday
1
1
Note: a total derivative would be something like $$frac{dw}{dt}=frac{partial w}{partial x}frac{dx}{dt}+frac{partial w}{partial y}frac{dy}{dt}+frac{partial w}{partial z}frac{dz}{dt}$$
– John Doe
yesterday
Note: a total derivative would be something like $$frac{dw}{dt}=frac{partial w}{partial x}frac{dx}{dt}+frac{partial w}{partial y}frac{dy}{dt}+frac{partial w}{partial z}frac{dz}{dt}$$
– John Doe
yesterday
See mixed partial derivatives
– WaveX
yesterday
See mixed partial derivatives
– WaveX
yesterday
Your notation with the "total derivative" makes absolutely no sense. I assume you mean $frac d{dt} f(x(t),y(t),z(t))$ or, often, $frac d{dt} f(t,x(t),y(t),z(t))$.
– Ted Shifrin
yesterday
Your notation with the "total derivative" makes absolutely no sense. I assume you mean $frac d{dt} f(x(t),y(t),z(t))$ or, often, $frac d{dt} f(t,x(t),y(t),z(t))$.
– Ted Shifrin
yesterday
add a comment |
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What I think you mean is (for example), something like this $$frac{partial}{partial x}frac{partial}{partial y}w$$
This is usually denoted by $$frac{partial^2 w}{partial xpartial y}$$and is defined by $$lim_{delta xto 0}lim_{delta y to 0}left(frac{f(x+delta x,y+delta y,z)-f(x+delta x,y,z)-f(x+delta x,y,z)+f(x,y,z)}{delta xdelta y}right)$$
As an example, if we let $f(x,y,z)=x^2y^3$, then $$frac{partial^2 f}{partial xpartial y}=frac{partial }{partial x}frac{partial }{partial y}(x^2y^3)=frac{partial }{partial x}(3x^2y^2)=6xy^2$$
answered yesterday
John Doe
10.4k11135
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What I think you mean is (for example), something like this $$frac{partial}{partial x}frac{partial}{partial y}w$$
This is usually denoted by $$frac{partial^2 w}{partial xpartial y}$$and is defined by $$lim_{delta xto 0}lim_{delta y to 0}left(frac{f(x+delta x,y+delta y,z)-f(x+delta x,y,z)-f(x+delta x,y,z)+f(x,y,z)}{delta xdelta y}right)$$
As an example, if we let $f(x,y,z)=x^2y^3$, then $$frac{partial^2 f}{partial xpartial y}=frac{partial }{partial x}frac{partial }{partial y}(x^2y^3)=frac{partial }{partial x}(3x^2y^2)=6xy^2$$
answered yesterday
John Doe
10.4k11135
add a comment |
What I think you mean is (for example), something like this $$frac{partial}{partial x}frac{partial}{partial y}w$$
This is usually denoted by $$frac{partial^2 w}{partial xpartial y}$$and is defined by $$lim_{delta xto 0}lim_{delta y to 0}left(frac{f(x+delta x,y+delta y,z)-f(x+delta x,y,z)-f(x+delta x,y,z)+f(x,y,z)}{delta xdelta y}right)$$
As an example, if we let $f(x,y,z)=x^2y^3$, then $$frac{partial^2 f}{partial xpartial y}=frac{partial }{partial x}frac{partial }{partial y}(x^2y^3)=frac{partial }{partial x}(3x^2y^2)=6xy^2$$
answered yesterday
John Doe
10.4k11135
add a comment |
What I think you mean is (for example), something like this $$frac{partial}{partial x}frac{partial}{partial y}w$$
This is usually denoted by $$frac{partial^2 w}{partial xpartial y}$$and is defined by $$lim_{delta xto 0}lim_{delta y to 0}left(frac{f(x+delta x,y+delta y,z)-f(x+delta x,y,z)-f(x+delta x,y,z)+f(x,y,z)}{delta xdelta y}right)$$
As an example, if we let $f(x,y,z)=x^2y^3$, then $$frac{partial^2 f}{partial xpartial y}=frac{partial }{partial x}frac{partial }{partial y}(x^2y^3)=frac{partial }{partial x}(3x^2y^2)=6xy^2$$
answered yesterday
John Doe
10.4k11135
What I think you mean is (for example), something like this $$frac{partial}{partial x}frac{partial}{partial y}w$$
This is usually denoted by $$frac{partial^2 w}{partial xpartial y}$$and is defined by $$lim_{delta xto 0}lim_{delta y to 0}left(frac{f(x+delta x,y+delta y,z)-f(x+delta x,y,z)-f(x+delta x,y,z)+f(x,y,z)}{delta xdelta y}right)$$
As an example, if we let $f(x,y,z)=x^2y^3$, then $$frac{partial^2 f}{partial xpartial y}=frac{partial }{partial x}frac{partial }{partial y}(x^2y^3)=frac{partial }{partial x}(3x^2y^2)=6xy^2$$
answered yesterday
John Doe
10.4k11135
answered yesterday
John Doe
10.4k11135
answered yesterday
John Doe
10.4k11135
answered yesterday
John Doe
10.4k11135
10.4k11135
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1
Note: a total derivative would be something like $$frac{dw}{dt}=frac{partial w}{partial x}frac{dx}{dt}+frac{partial w}{partial y}frac{dy}{dt}+frac{partial w}{partial z}frac{dz}{dt}$$
– John Doe
yesterday
See mixed partial derivatives
– WaveX
yesterday
Your notation with the "total derivative" makes absolutely no sense. I assume you mean $frac d{dt} f(x(t),y(t),z(t))$ or, often, $frac d{dt} f(t,x(t),y(t),z(t))$.
– Ted Shifrin
yesterday