3.19.5 Tensor Covector Gradient Interpretation

Understanding how tensor covector gradients operate in multilinear algebra and their role in differential geometry and physical applications.

Tensor Covector Gradient Interpretation is the identification of a covector with the differential of a scalar-valued function, the reading under which a covector at a point of a manifold, or on a vector space regarded as its own tangent space, represents the linear approximation of a function's rate of change, distinct from the gradient vector obtained only after a metric has been introduced. This interpretation is the primary bridge between elementary multivariable calculus, where gradients are typically drawn as arrows, and the covector formalism of tensor algebra, where the differential is correctly recognized as a dual object requiring no inner product.

The Differential as a Covector

Definition of the Differential

For a smooth function f defined near a point p of a manifold M, the differential df_p is the linear functional on the tangent space T_pM defined by the directional derivative of f along each tangent vector.

d f_{p} (v) = lim_{t \to 0} \frac{f (γ (t)) - f (p)}{t}

where gamma is any curve through p with gamma(0) = p and velocity gamma'(0) = v. Linearity of df_p in v follows from the ordinary rules of differentiation, and df_p is by construction an element of the cotangent space T_p*M, the dual of the tangent space, making it a covector rather than a vector from the outset.

Coordinate Expression

In local coordinates x^1, ..., x^n, the differential expands in the coordinate dual basis dx^1, ..., dx^n as the familiar sum of partial derivatives.

d f = \sum_{i = 1}^{n} \frac{\partial f}{\partial x^{i}} d x^{i}

The coefficients partial f over partial x^i are the components of the covector df in the coordinate dual basis, and this expression is the origin of the classical notation dx, dy, dz for the dual basis covectors dual to the coordinate vector fields.

Distinguishing the Differential from the Gradient Vector

The Gradient Requires a Metric

The gradient vector, familiar from multivariable calculus as an arrow pointing in the direction of steepest ascent, is obtained from the differential only after choosing an inner product g on the tangent space, via the musical isomorphism raising the index of df.

grad f = (^{♯)} (d f)

so that g(grad f, v) equals df(v) for every tangent vector v. Since this construction depends on g, different choices of metric on the same manifold yield different gradient vectors from the same differential, while the differential df itself is canonical and independent of any metric, existing as long as f is differentiable. The gradient interpretation of a covector therefore properly refers to df, not to grad f, and treating the two as interchangeable is only valid once a specific metric has been fixed.

Level Sets and Steepest Ascent

Under the gradient interpretation, the kernel of df_p is the tangent hyperplane to the level set of f through p, and the covector df_p, viewed as a hyperplane stack, has its hyperplanes most tightly packed in the directions where f changes fastest. This recovers the classical intuition of closely spaced contour lines indicating steep terrain, but expressed without reference to any arrow or magnitude, purely in terms of the spacing of the level sets of the linear approximation df_p.

Behavior Under Composition and Pullback

The Chain Rule as Pullback

If phi is a smooth map from a manifold N to a manifold M and f is a smooth function on M, the differential of the composite function f composed with phi equals the pullback, via the differential of phi, of the differential of f.

d (f \circ φ) = {(d φ_{q})}^{*} (d f_{φ (q)})

for q in N, where d phi_q is the differential, a linear map on tangent spaces, of phi at q. This identity is precisely the chain rule of multivariable calculus, restated as a statement about pullback of covectors: the gradient interpretation reveals the chain rule to be nothing more than the naturality of the pullback operation applied to the differential of a function.

Gradients Do Not Push Forward

Because the differential is a covector, it pulls back along phi rather than pushing forward, and this asymmetry is exactly why the chain rule composes in the order it does, with d phi_q acting first on tangent vectors and its pullback acting oppositely on covectors. A gradient vector, being tied to a choice of metric, does not transform this simply under phi unless phi is compatible with the metrics on both manifolds, for example an isometry; the differential, by contrast, transforms correctly under an arbitrary smooth map, which is the fundamental reason the gradient interpretation of a covector is considered more primitive than the gradient-vector picture inherited from calculus.

Higher-Order and Iterated Differentials

Differentials of Differentials

Applying the exterior derivative twice to a function f always yields zero, an identity usually written d(df) = 0, which reflects the fact that the covector field df, arising as a gradient-type object, is automatically closed. This closedness property is special to covectors that arise as differentials of functions, called exact covector fields, and it distinguishes them from general covector fields, which need not satisfy any such integrability condition, marking the gradient interpretation as a genuinely restrictive special case within the broader theory of covector fields on a manifold.