![]() In the literature, several independent definitions and theories of nonlocal and fractional vector. However, it is not clear to me how the second term in my expression can be rearranged to arrive at what the author is showing.Īm I doing something wrong or is there simply an extra step required that I am not seeing? For reference, the paper that I am referring to is "Physical, Mathematical, and Numerical Derivations of the Cahn-Hilliard Equation" by Lee, Huh, Jeong, Shin, Yun, and Kim. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. This identity can be established by use of. $$ =\int_\Omega \mu M\nabla\cdot\nabla\mu\ d\vec=0$ on $\partial\Omega$). where, in the surface integral on the right-hand side, n denotes the outward-drawn normal at any given point on. If I make this substitution in the second line, I then have that: ![]() ![]() $$ \nabla \cdot (M\nabla\mu) = M\nabla\cdot\nabla\mu+\nabla\mu\cdot\nabla M $$ Using vector identities again, for a scalar function $M(c)$ and a vector function $\nabla\mu$, we have that: However, I don't quite see how to then arrive at the third line. The traditional topics are covered: basic vector algebra lines, planes and surfaces vector-valued functions functions of 2 or 3 variables partial derivatives optimization multiple integrals line and surface integrals. So everything makes sense to me up until that second line. This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. To get to the second line, the author then uses the fact that: Using this, I can arrive at the rightmost expression on the first line. $$ \nabla c \cdot \nabla c_t = \nabla\cdot c_t\nabla c-c_t \Delta c $$ 1 Vectors in Euclidean Space 2 Functions of Several Variables 3 Multiple Integrals 4 Line and Surface Integrals Ancillary Material Michael Corral About the Book This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. In the first line, the author uses the definition of $\mu = F'(c) - \epsilon^2\Delta c$ and then the vector identity that states: I have been trying to follow along with how the author went from line 2 to line 3. The question I have is related to the following section of the paper: We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. ![]() I am trying to follow along a review paper regarding the derivation of the Cahn-Hilliard paper. ![]()
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