- #1

junt

- 18

- 1

Hi All,

$$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$

I can perform the integration of the integral above easily by changing the variable

$$u=x_2+x_1\\

v=x_2-x_1$$

Of course first computing the Jacobian, and integrating over ##u## and ##v##

I am wondering how you perform the change of variable for 4-dimensional integral like below:

$$\int{\exp(\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i))dx_1dx_2dx_3dx_4}$$

Is it something like:

$$x_2-x_1=u \\

x_2+x_1=v \\

x_4-x_3=p \\

x_4+x_3=q$$

Should this be enough? I think one only needs 4 new variables right? Because I was thinking the integral would be easier if I could do something like:

$$x_2-x_1=u \\

x_3-x_2=v \\

x_4-x_3=p \\

x_1-x_4=q$$

Or in general, how do you perform change of variable in multi-dimensional case. Should you generally perform it by doing:

$$u=a_1x_1+a_2x_2+a_3x_3+a_4x_4 \\

v=b_1x_1+b_2x_2+b_3x_3+b_4x_4 \\

p=c_1x_1+c_2x_2+c_3x_3+c_4x_4 \\

q=d_1x_1+d_2x_2+d_3x_3+d_4x_4$$

And what kind of change of variable does one need to perform to perform the integration of such integration easily?

$$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$

I can perform the integration of the integral above easily by changing the variable

$$u=x_2+x_1\\

v=x_2-x_1$$

Of course first computing the Jacobian, and integrating over ##u## and ##v##

I am wondering how you perform the change of variable for 4-dimensional integral like below:

$$\int{\exp(\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i))dx_1dx_2dx_3dx_4}$$

Is it something like:

$$x_2-x_1=u \\

x_2+x_1=v \\

x_4-x_3=p \\

x_4+x_3=q$$

Should this be enough? I think one only needs 4 new variables right? Because I was thinking the integral would be easier if I could do something like:

$$x_2-x_1=u \\

x_3-x_2=v \\

x_4-x_3=p \\

x_1-x_4=q$$

Or in general, how do you perform change of variable in multi-dimensional case. Should you generally perform it by doing:

$$u=a_1x_1+a_2x_2+a_3x_3+a_4x_4 \\

v=b_1x_1+b_2x_2+b_3x_3+b_4x_4 \\

p=c_1x_1+c_2x_2+c_3x_3+c_4x_4 \\

q=d_1x_1+d_2x_2+d_3x_3+d_4x_4$$

And what kind of change of variable does one need to perform to perform the integration of such integration easily?

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