Question

Consider a two dimensional system where the vector \vec{r} is given by \vec{r} = x_1 \vec{e_1} + x_2 \vec{e_2} If the coordinate is transformed into a different, non-orthogonal coordinates as $x'_1 = \frac{1}{\sqrt{2}}(x_1 + x_2)$ $x'_2 = x_2$ The gradient of the scalar function is: $\nabla \phi(\vec{x}) = \partial_1 \phi \vec{e_1} + \partial_2 \phi \vec{e_2}$ Find the components of the gradient in the primed co-ordinates and show that it transforms as a covariant vector.

          Consider a two dimensional system where the vector \vec{r} is given by
\vec{r} = x_1 \vec{e_1} + x_2 \vec{e_2}
If the coordinate is transformed into a different, non-orthogonal coordinates as
$x'_1 = \frac{1}{\sqrt{2}}(x_1 + x_2)$
$x'_2 = x_2$
The gradient of the scalar function is:
$\nabla \phi(\vec{x}) = \partial_1 \phi \vec{e_1} + \partial_2 \phi \vec{e_2}$
Find the components of the gradient in the primed co-ordinates and show that it
transforms as a covariant vector.

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Consider a two dimensional system where the vector r⃗ is given by
r⃗ = x1 e⃗1⃗ + x2 e⃗2⃗
If the coordinate is transformed into a different, non-orthogonal coordinates as
x'1 = (1)/(√(2))(x1 + x2)
x'2 = x2
The gradient of the scalar function is:
∇ϕ(x⃗) = ∂1 ϕe⃗1⃗ + ∂2 ϕe⃗2⃗
Find the components of the gradient in the primed co-ordinates and show that it
transforms as a covariant vector.

Added by Greg -.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Zhumagali Shomanov

05/28/2022

Step 1

In this case, the gradient is given by: ∇f = (∂f/∂x₁)ê₁ + (∂f/∂x₂)ê₂ where ∂f/∂x₁ and ∂f/∂x₂ are the partial derivatives of the scalar function f with respect to x₁ and x₂, respectively. Show more…

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re not 5. Consider a two dimensional system where the vector is given by T = x₁ê₁ + x₂ê2 and the x-coordinate is transformed into a different, non-orthogonal coordinates as: x₁ x₂ - The gradient of the scalar function is: √2 X2 (x1 + x₂) (x) = a₁0₁ + 0₂0 êz Find the components of the gradient in the primed co-ordinates and show that it transforms as a covariant vector. V=+02e2 transforms as a covariant vector.

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