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Find the Jacobian of the transformation. x=6u+−v,y=−5u+−5v ∂(u,v)∂(x,y)​=

User Neobie
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1 Answer

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The Jacobian of a transformation measures how the variables in the original coordinate system change when transformed into a new coordinate system. In this case, we have the transformation:

x = 6u - v
y = -5u - 5v

To find the Jacobian of this transformation, we need to calculate the partial derivatives of x with respect to u and v, and the partial derivatives of y with respect to u and v. The Jacobian matrix is then formed by arranging these partial derivatives in a specific order.

Let's calculate the partial derivatives first:

∂x/∂u = 6
∂x/∂v = -1
∂y/∂u = -5
∂y/∂v = -5

Now we can form the Jacobian matrix:

J = [∂x/∂u ∂x/∂v]
[∂y/∂u ∂y/∂v]

Substituting the partial derivatives we calculated earlier, we get:

J = [6 -1]
[-5 -5]

So, the Jacobian of the transformation x = 6u - v and y = -5u - 5v is:

J = [6 -1]
[-5 -5]

User Kemen Paulos Plaza
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