Suppose that f is a function of two variables (y and z) only.!Show that the gradient ∇ f = (∂ f /∂ y)ˆy + (∂ f /∂z)ˆz transforms as a vector un-der rotations, Eq. 1.29. [Hint: (∂ f /∂ y) = (∂ f /∂ y)(∂ y/∂ y) + (∂ f /∂z)(∂z/∂ y),and the analogous formula for ∂ f /∂z. We know that y = y cos φ + z sin φ andz = −y sin φ + z cos φ; “solve” these equations for y and z (as functions of yand z), and compute the needed derivatives ∂ y/∂ y, ∂z/∂ y, etc.]
Question
Suppose that f is a function of two variables (y and z) only.!Show that the gradient ∇ f = (∂ f /∂ y)ˆy + (∂ f /∂z)ˆz transforms as a vector un-der rotations, Eq. 1.29. [Hint: (∂ f /∂ y) = (∂ f /∂ y)(∂ y/∂ y) + (∂ f /∂z)(∂z/∂ y),and the analogous formula for ∂ f /∂z. We know that y = y cos φ + z sin φ andz = −y sin φ + z cos φ; “solve” these equations for y and z (as functions of yand z), and compute the needed derivatives ∂ y/∂ y, ∂z/∂ y, etc.]
Solution
To show that the gradient ∇f = (∂f/∂y)ˆy + (∂f/∂z)ˆz transforms as a vector under rotations, we need to use the given hint and the known equations for y and z in terms of φ.
First, let's write down the equations for y and z:
y = y cos φ + z sin φ z = -y sin φ + z cos φ
We can "solve" these equations for y and z as functions of y and z by differentiating both sides of each equation with respect to y and z. This gives us the needed derivatives ∂y/∂y, ∂z/∂y, etc.
Differentiating the first equation with respect to y gives:
∂y/∂y = cos φ + z(∂ sin φ/∂y)
And differentiating the first equation with respect to z gives:
∂y/∂z = y(∂ cos φ/∂z) + sin φ
Similarly, differentiating the second equation with respect to y gives:
∂z/∂y = -sin φ + z(∂ cos φ/∂y)
And differentiating the second equation with respect to z gives:
∂z/∂z = -y(∂ sin φ/∂z) + cos φ
Now, we can substitute these derivatives into the hint equation:
(∂f/∂y) = (∂f/∂y)(∂y/∂y) + (∂f/∂z)(∂z/∂y)
This gives us:
(∂f/∂y) = (∂f/∂y)(cos φ + z(∂ sin φ/∂y)) + (∂f/∂z)(-sin φ + z(∂ cos φ/∂y))
And similarly for ∂f/∂z:
(∂f/∂z) = (∂f/∂y)(y(∂ cos φ/∂z) + sin φ) + (∂f/∂z)(-y(∂ sin φ/∂z) + cos φ)
These equations show that the gradient ∇f transforms as a vector under rotations, as required.
Similar Questions
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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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