Covariant Nature of the Gradient | Proof and Derivation

The derivation proves that the partial derivatives of a scalar field, $\partial_a \phi$, naturally form the covariant components of a vector. This is a fundamental concept in tensor calculus because a scalar field's value is independent of the coordinate system. By applying the chain rule to a coordinate transformation, the partial derivatives are shown to transform in a manner identical to the definition of a covariant vector. This means the transformation rule for a covariant vector, $V_b^{\prime}= \sum_a \frac{\partial x^a}{\partial x^b} V_a$, perfectly matches the transformation of the partial derivatives, $\frac{\partial \phi^{\prime}}{\partial x^b}=\sum_a \frac{\partial \phi}{\partial x^a} \frac{\partial x^a}{\partial x^b}$. This result validates that the gradient, which is a vector composed of these partial derivatives, is a quintessential example of a covariant vector.

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✍️Mathematical Proof

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A scalar field is coordinate-independent. The core principle is that the value of a scalar quantity, such as temperature or pressure, remains the same regardless of the coordinate system used to describe it. This is expressed as $\phi\left(x^a\right)=\phi^{\prime}\left(x^{\prime b}\right)$.

The transformation of a covariant vector is defined by a specific rule. A vector with covariant components, $V_a$, transforms to a new set of components, $V_b^{\prime}$, using the Jacobian matrix of the coordinate transformation: $V_b^{\prime}=\sum_a \frac{\partial x^a}{\partial x^b} V_a$.

Partial derivatives of a scalar field naturally follow this rule. By applying the chain rule, the transformation of partial derivatives of the scalar field is found to be $\frac{\partial \phi^{\prime}}{\partial x^b}=\sum_a \frac{\partial \phi}{\partial x^a} \frac{\partial x^a}{\partial x^b}$.

The gradient is a covariant vector. Because the transformation rule for the partial derivatives of a scalar field is identical to the transformation rule for a covariant vector, the partial derivatives ( $\partial_a \phi$ ) are formally identified as the covariant components of a vector. This is why the gradient, which is the vector of all partial derivatives, is a fundamental example of a covariant vector.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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