Mixing
Where is the following $Bbb R^3 rightarrow Bbb R$ function continuously differentiable?
Clash Royale CLAN TAG#URR8PPP
up vote
-1
down vote
favorite
Where is $f(x,y,z)=begincases
0 & text$xyx=0$ \
x^2y^2z^2sin big (frac1xyz big) & text$xyz ne 0$ endcases$ continuously differentiable?
What I got so far is that $f$ can be written as a composition, $f=hcirc g$ where $g(x,y,z)=xyz$ and $h(t)=t^2sin big(frac1t big)$. By the chain rule we have that: $nabla f(x,y,z) = h'(g(x,y,z))cdot nabla g(x,y,z) = big (2xyzcdot sin big( frac1xyz big) - cos big( frac1xyz big)big)cdot (yz,xz,xy)$.
Is that even correct? What should the gradient be at the origin or in other points where $xyz=0$? I'm really not sure how this fits into the mix. Also, is the function continuously differentiable in these points?
multivariable-calculus derivatives
asked Jul 19 at 16:40
Noa
529514
add a comment |Â
up vote
-1
down vote
favorite
Where is $f(x,y,z)=begincases
0 & text$xyx=0$ \
x^2y^2z^2sin big (frac1xyz big) & text$xyz ne 0$ endcases$ continuously differentiable?
What I got so far is that $f$ can be written as a composition, $f=hcirc g$ where $g(x,y,z)=xyz$ and $h(t)=t^2sin big(frac1t big)$. By the chain rule we have that: $nabla f(x,y,z) = h'(g(x,y,z))cdot nabla g(x,y,z) = big (2xyzcdot sin big( frac1xyz big) - cos big( frac1xyz big)big)cdot (yz,xz,xy)$.
Is that even correct? What should the gradient be at the origin or in other points where $xyz=0$? I'm really not sure how this fits into the mix. Also, is the function continuously differentiable in these points?
multivariable-calculus derivatives
asked Jul 19 at 16:40
Noa
529514
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Where is $f(x,y,z)=begincases
0 & text$xyx=0$ \
x^2y^2z^2sin big (frac1xyz big) & text$xyz ne 0$ endcases$ continuously differentiable?
What I got so far is that $f$ can be written as a composition, $f=hcirc g$ where $g(x,y,z)=xyz$ and $h(t)=t^2sin big(frac1t big)$. By the chain rule we have that: $nabla f(x,y,z) = h'(g(x,y,z))cdot nabla g(x,y,z) = big (2xyzcdot sin big( frac1xyz big) - cos big( frac1xyz big)big)cdot (yz,xz,xy)$.
Is that even correct? What should the gradient be at the origin or in other points where $xyz=0$? I'm really not sure how this fits into the mix. Also, is the function continuously differentiable in these points?
multivariable-calculus derivatives
asked Jul 19 at 16:40
Noa
529514
Where is $f(x,y,z)=begincases
0 & text$xyx=0$ \
x^2y^2z^2sin big (frac1xyz big) & text$xyz ne 0$ endcases$ continuously differentiable?
What I got so far is that $f$ can be written as a composition, $f=hcirc g$ where $g(x,y,z)=xyz$ and $h(t)=t^2sin big(frac1t big)$. By the chain rule we have that: $nabla f(x,y,z) = h'(g(x,y,z))cdot nabla g(x,y,z) = big (2xyzcdot sin big( frac1xyz big) - cos big( frac1xyz big)big)cdot (yz,xz,xy)$.
Is that even correct? What should the gradient be at the origin or in other points where $xyz=0$? I'm really not sure how this fits into the mix. Also, is the function continuously differentiable in these points?
multivariable-calculus derivatives
asked Jul 19 at 16:40
Noa
529514
edited Jul 20 at 6:19
asked Jul 19 at 16:40
Noa
529514
asked Jul 19 at 16:40
Noa
529514
asked Jul 19 at 16:40
Noa
529514
529514
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
Yes, what you have is correct.
Now you need to consider the case, where
$$x = 0 quad lor quad y = 0 quad lor quad z = 0,$$
and then consider the function
$$f = h circ g.$$
After finding $Df$ piecewisely, you need to check whether that piecewise function, $Df$, is continuous or not.
answered Jul 20 at 6:29
onurcanbektas
3,0611834
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Yes, what you have is correct.
Now you need to consider the case, where
$$x = 0 quad lor quad y = 0 quad lor quad z = 0,$$
and then consider the function
$$f = h circ g.$$
After finding $Df$ piecewisely, you need to check whether that piecewise function, $Df$, is continuous or not.
answered Jul 20 at 6:29
onurcanbektas
3,0611834
add a comment |Â
up vote
0
down vote
Yes, what you have is correct.
Now you need to consider the case, where
$$x = 0 quad lor quad y = 0 quad lor quad z = 0,$$
and then consider the function
$$f = h circ g.$$
After finding $Df$ piecewisely, you need to check whether that piecewise function, $Df$, is continuous or not.
answered Jul 20 at 6:29
onurcanbektas
3,0611834
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Yes, what you have is correct.
Now you need to consider the case, where
$$x = 0 quad lor quad y = 0 quad lor quad z = 0,$$
and then consider the function
$$f = h circ g.$$
After finding $Df$ piecewisely, you need to check whether that piecewise function, $Df$, is continuous or not.
answered Jul 20 at 6:29
onurcanbektas
3,0611834
Yes, what you have is correct.
Now you need to consider the case, where
$$x = 0 quad lor quad y = 0 quad lor quad z = 0,$$
and then consider the function
$$f = h circ g.$$
After finding $Df$ piecewisely, you need to check whether that piecewise function, $Df$, is continuous or not.
answered Jul 20 at 6:29
onurcanbektas
3,0611834
answered Jul 20 at 6:29
onurcanbektas
3,0611834
answered Jul 20 at 6:29
onurcanbektas
3,0611834
answered Jul 20 at 6:29
onurcanbektas
3,0611834
3,0611834
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2856823%2fwhere-is-the-following-bbb-r3-rightarrow-bbb-r-function-continuously-diffe%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password