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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. So everything makes sense to me up until that second line. To get to the second line, the author then uses the fact that: Similarly to regular calculus, matrix and vector calculus rely on a set of identities to make computations more manageable. Using this, I can arrive at the rightmost expression on the first line. $$ \nabla c \cdot \nabla c_t = \nabla\cdot c_t\nabla c-c_t \Delta c $$ 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: (This is a stronger condition than having k derivatives, as shown by the second example of Smoothness § Examples.I am trying to follow along a review paper regarding the derivation of the Cahn-Hilliard paper. If in addition the kth derivative is continuous, then the function is said to be of differentiability class C k. A function that has k successive derivatives is called k times differentiable. Similar examples show that a function can have a kth derivative for each non-negative integer k but not a ( k + 1)th derivative. DefinitionĪ function of a real variable f( x) is differentiable at a point a of its domain, if its domain contains an open interval I containing a, and the limit L = lim h → 0 f ( a + h ) − f ( a ) h, and it does not have a derivative at zero. Differentiation and integration constitute the two fundamental operations in single-variable calculus. The fundamental theorem of calculus relates antidifferentiation with integration. The reverse process is called antidifferentiation. The process of finding a derivative is called differentiation. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. It can be calculated in terms of the partial derivatives with respect to the independent variables. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. Replace every x x with a y y and replace every y y with an x x. This is done to make the rest of the process easier. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. Given the function f (x) f ( x) we want to find the inverse function, f 1(x) f 1 ( x). 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For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. Derivatives are a fundamental tool of calculus. Let’s first identify \ (P\) and \ (Q\) and then check that the vector field is conservative. Determine if the following vector fields are conservative and find a potential function for the vector field if it is conservative. Vectors provide a simple way to write down an equation to determine the position. It is usually best to see how we use these two facts to find a potential function in an example or two. In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Math Input Calculus & Sums More than just an online integral solver. |
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