Jacobian matrices

  • #1
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[tex]x=t^2-s^2, y=ts,u=x,v=-y[/tex]

a) compute derivative matrices


[tex]\vec{D}f(x,y) = \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right][/tex]

[tex]\vec{D}f(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right][/tex]

b) express (u,v) in terms of (t,s)

[tex]f(u(x,y),v(x,y) = (t^2-s^2,-(ts))[/tex]

c) Evaluate [tex]\vec{D}(u,v)[/tex]

[tex]\vec{D}(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right][/tex]

[tex]= \left[\begin{array}{cc}2t&-2s\\-s&-t\end{array}\right][/tex]

d) verify if chain rule holds


need help with this last part, also need to know if I even did the rest correctly
 
Last edited:

Answers and Replies

  • #2
STEMucator
Homework Helper
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[tex]x=t^2-s^2, y=ts,u=x,v=-y[/tex]

a) compute derivative matrices


[tex]\vec{D}f(x,y) = \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right][/tex]

[tex]\vec{D}f(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right][/tex]

b) express (u,v) in terms of (t,s)

[tex]f(u(x,y),v(x,y) = (t^2-s2,-(ts))[/tex]

c) Evaluate [tex]\vec{D}(u,v)[/tex]

[tex]\vec{D}(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right][/tex]

[tex]= \left[\begin{array}{cc}2t&-2s\\-s&-t\end{array}\right][/tex]

d) verify if chain rule holds


need help with this last part, also need to know if I even did the rest correctly
Part a) looks okay, but the notation is a little rough. Something more like:

[tex]
J_{s, t}(x,y) = \begin{pmatrix}
x_s & x_t \\
y_s & y_t
\end{pmatrix}, \quad

J_{x, y}(u,v) = \begin{pmatrix}
u_x & u_y \\
v_x & v_y
\end{pmatrix}
[/tex]

##J_{s, t}(x,y)## and ##J_{x, y}(u,v)## are the Jacobian matricies.

I'm sure you meant ##u = t^2 - s^2## for part b).

For part c), I'm sure what is intended is you find the derivative matrix ##J_{s, t}(u,v)## after expressing ##u## and ##v## as functions of ##s## and ##t##.
 
  • #3
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Part a) looks okay, but the notation is a little rough. Something more like:

[tex]
J_{s, t}(x,y) = \begin{pmatrix}
x_s & x_t \\
y_s & y_t
\end{pmatrix}, \quad

J_{x, y}(u,v) = \begin{pmatrix}
u_x & u_y \\
v_x & v_y
\end{pmatrix}
[/tex]

##J_{s, t}(x,y)## and ##J_{x, y}(u,v)## are the Jacobian matricies.

I'm sure you meant ##u = t^2 - s^2## for part b).

For part c), I'm sure what is intended is you find the derivative matrix ##J_{s, t}(u,v)## after expressing ##u## and ##v## as functions of ##s## and ##t##.
I fixed the error in (b) and yeah thats what I want for (c), I'm using the notation that is used in my book/what my professor uses

as for part (d) how do I verify this
 
  • #4
STEMucator
Homework Helper
2,075
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I fixed the error in (b) and yeah thats what I want for (c), I'm using the notation that is used in my book/what my professor uses

as for part (d) how do I verify this
For part d), if ##f## is a function of ##u## and ##v##, which are functions of ##x## and ##y##, which are functions of ##s## and ##t##, what is the partial derivative of ##f## with respect to ##u##? How about the partial with respect to ##v##?
 
  • #5
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0
[tex] (\frac{\partial f}{\partial u})_{t,s} = 2t,-2s \,\,\, (\frac{\partial f}{\partial v})_{t,s} = -s,-t [/tex]

I know this is kind of an abuse of notation but I've been typing LaTeX all day
 
Last edited:
  • #6
STEMucator
Homework Helper
2,075
140
[tex] (\frac{\partial f}{\partial u})_{t,s} = 2t,-2s \,\,\, (\frac{\partial f}{\partial v})_{t,s} = -s,-t [/tex]

I know this is kind of an abuse of notation but I've been typing LaTeX all day
Looks good.
 
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