Lecture 06: Derivatives and Gradients
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Overview
Steven M. Wise develops a coordinate-independent framework for differentiating scalar and vector fields in continuum mechanics. He introduces norm-based big-O and little-o estimates, defines the scalar gradient and vector-field gradient through Fréchet differentiability, derives their component formulas, and begins defining divergence as the trace of the vector-field gradient.
Key takeaways
- Fréchet differentiability expresses a scalar field locally as φ(x + h) = φ(x) + ∇φ(x) · h + o(||h||), defining the gradient without choosing coordinates.
- The directional derivative in direction a equals ∇φ(x) · a, but existence of directional derivatives alone is weaker than Fréchet differentiability.
- In an orthonormal frame, the scalar gradient is ∇φ = φ_,i e_i, while the vector-field gradient is ∇v = v_i,j e_i ⊗ e_j.
- A vector-field gradient is a second-order tensor, and its trace gives divergence: div v = v_i,i.
- If a scalar field has a Lipschitz continuous gradient, its first-order Taylor approximation has a quadratic error, O(||h||^2), rather than only an o(||h||) remainder.
Chapters
- Wise distinguishes localized points in E3 from movable vectors, while identifying them when convenient for calculations.
- The Euclidean vector norm is the square root of a vector's dot product with itself.
- A second-order tensor can use the square root of its tensor inner product with itself, giving the Frobenius norm rather than an induced operator norm.
- Big-O of order r bounds the norm of f(u) by C times the norm of u raised to r, for 0 < ||u|| < delta.
- Little-o of order r means ||f(u)|| / ||u||^r tends to zero, so f(u) vanishes faster than ||u||^r.
- The scalar example x^2 / x = x illustrates why x^2 tends to zero faster than x.
- Little-o of ||u||^r always implies big-O of ||u||^r, but big-O of the same order does not imply little-o.
- If p > r and f is big-O of ||u||^p, then f is little-o of ||u||^r.
- The shorthand f = g + O(u^r) means f - g is O(u^r), with the analogous convention for little-o.
- A scalar field maps points in an open or specified region of E3 to real numbers; pressure and temperature are examples.
- A vector field maps points to vectors, as with ocean-current velocity, gravitational force, or elastic displacement.
- A tensor field maps points to second-order tensors, with stress and strain tensors as central continuum-mechanics examples.
- Wise shifts from classifying fields to differentiating them, starting with scalar fields as the simplest case.
- The scalar differentiability definition requires an open domain so small perturbations around a point remain in the domain.
- The lecture materials are intended to be available through GitHub, and Wise notes that some handwritten notes still need syncing.
- At x, a scalar field φ is Fréchet differentiable if φ(x + h) = φ(x) + g · h + o(||h||) as h tends to zero.
- The vector g is defined as the gradient of φ at x and provides the linear part of the local change in the field.
- The little-o remainder generalizes the one-dimensional difference-quotient idea to vector inputs.
- Fréchet differentiability implies the directional derivative formula, but the directional or Gâteaux condition is weaker and does not by itself guarantee Fréchet differentiability.
- For a direction a, the directional derivative is obtained by differentiating φ(x + s a) with respect to s at s = 0.
- The directional change is scaled by the length of a; a unit vector gives the derivative per unit distance in that direction.
- In an orthonormal frame, Wise writes a direction as a = a_i e_i and applies the multivariable chain rule to φ(x + s a).
- Differentiating each coordinate x_i + s a_i with respect to s produces the component a_i.
- The resulting directional derivative is φ_,i a_i, with the repeated index summed from 1 to 3.
- The component expression φ_,i a_i can be rewritten as (φ_,i e_i) · a using the orthonormal-frame dot product.
- Because a is arbitrary, equality of dot products with every a implies equality of the vectors, yielding ∇φ = φ_,i e_i.
- This component formula matches the familiar calculus gradient while the definition itself remains frame invariant.
- Taking the gradient of a scalar field at each point produces a vector field; the gradient of a physical potential can represent force in a conservative setting.
- Wise assumes fields are smooth enough that derivatives of every order being used exist and are continuous.
- The notation φ_,i denotes differentiation with respect to x_i, and repeated indices imply summation; commas are reserved for derivatives in continuum mechanics.
- For a C1 scalar field whose gradient is Lipschitz with constant γ, gradient differences are bounded by γ times the distance between points.
- Taylor's theorem then gives a first-order approximation φ(x + h) = φ(x) + ∇φ(x) · h + O(||h||^2).
- The quadratic remainder is stronger than the basic Fréchet definition's o(||h||) remainder and supports more precise approximation estimates.
- A vector field v is Fréchet differentiable at x when a second-order tensor A gives v(x + h) = v(x) + A h + o(||h||).
- Wise calls A the gradient of the vector field; it is also commonly called the Jacobian matrix.
- As with scalar fields, the Fréchet condition implies the weaker directional derivative condition, but the converse need not hold.
- In an orthonormal frame, write v = v_i e_i and x = x_j e_j; applying the chain rule gives derivatives of each v_i with respect to each x_j.
- The tensor gradient has the component form ∇v = v_i,j e_i ⊗ e_j, with summation over both repeated indices.
- The free index i identifies the vector component, while j identifies the coordinate direction of differentiation.
- Evaluating the vector-field gradient at every point produces a second-order tensor field, increasing the number of components compared with the original vector field.
- Divergence is defined coordinate-independently as the trace of ∇v, reducing the second-order tensor to a scalar field.
- In an orthonormal frame, taking the trace gives div v = v_i,i, the familiar sum of each component's derivative with respect to its matching coordinate.
Transcript excerpt
0:05 All right, welcome back to continue mechanics. This is class number six. Today we're talking about tensor calculus is our first this our first lecture on tensor calculus after having finished um learning about tensor algebra and all we really need to know about that subject. We're going to talk about derivatives and functions uh importantly fields and a lot of stuff related to that. Of course there are a lot of different types of objects in tensor calculus and tensor algebra. So it's not hard to imagine that there are different types of functions that you could um describe
0:52 in this in this field and therefore there are different types of derivatives that we we will encounter. They're really the same in some sense but they have different actions as we'll see. Before we get into this we need to have some sort of uh notion about distances. Distances in vector spaces are usually given by a norm. Uh that's because you can often times put a norm on a vector space. And in our situation u that's the case. So we have the the standard arrows or vectors in threedimensional space and you can put the uklitian distance on those things.
1:39 So, um, we're not going to worry too much about again the distinction between points in three space and vectors in threespace because they can really be identified. The only, uh, distinction that you should keep in mind is that vectors can be picked up and moved as long as you don't change the magnitude and direction of them. They're always the same. But points can never be moved. They're uh, completely localized in space somewhere. So I'm not going to worry too much about this definition. It's the same as this one essentially. So it's defining the ukitian norm on a
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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.