If x=rcosθ , y=rsinθ then ∂r/∂x X ∂x/∂r=
Question
If x=rcosθ , y=rsinθ then ∂r/∂x X ∂x/∂r=
Solution
The given equations are in polar coordinates where r is the radial distance and θ is the angle.
First, let's find ∂r/∂x and ∂x/∂r.
From x = rcosθ, we can differentiate both sides with respect to x to get:
∂r/∂x = 1/(cosθ)
Similarly, differentiating both sides of the same equation with respect to r, we get:
∂x/∂r = cosθ
Now, we can find the product of ∂r/∂x and ∂x/∂r:
∂r/∂x * ∂x/∂r = 1/(cosθ) * cosθ = 1
So, ∂r/∂x * ∂x/∂r = 1.
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Partial Differentiation of Function of Several Variables. Differentiation of Implicit Functions, Theorem and Applications. Jacobians. Differentiation of a Vector Function of Several Variables. Total Differential, Tangent Plane to Surface, the Tangent Vector. Curvilinear Coordinates; Plane Polar, Cylindrical and Spherical Co-ordinates. Multiple Integrals. Line Integrals, Multiple Surface and Volume Integral. Gradient, Divergence and Curl. The Theorems of Green, Gauss, and Stokes
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