Differential calculus is used to find rates of change in the grade of a slope or a curve. Mathematicians typically use a combination of integral and differential calculus in solving their problems. Let's look at an example. Imagine that you need to push a box up a hill on a smooth, ...
I don't want to sound egotistical here, I don't think that I am beyond the knee of this curve and I don't know anyone who is. I do know some people that were. I think, for example, that anyone will agree that Isaac Newton would be well on the top of this curve. When you th...
Math Calculus Continuous functions What does a smooth embedding mean?Question:What does a smooth embedding mean?Functions in MathA function in mathematics is defined as the relation between two sets, these sets have corresponding values where every element or value for one of the set is ...
I think there is some ambiguity in the use of the word 'curve'. To me, a 'curve' in a manifold M is a smooth map γ:I→M (where I is some interval in R containg 0). You seem to be using 'curve' as the image of such a map, right? Feb 13, 2011 #8 HallsofIvy Scien...
So we need curvature, defined in Calculus, and is a function of position on the curve, which can be summarized as k(t)=|\gamma^{''}(t)|, where k(t) is the curvature of a plane curve \gamma. It also means the length of the acceleration vector, when \gamma is given a unit-...
In general, we can define order of contact of a line with a curve at a point (a, f(a)) by which power of x-a divides the difference between the equation of the line and the equation of the curve. A tangent line is one for which the order of contact is >= 2, and an infllec...
If f and g are differentiable functions of one variable, prove that \int _{C}f(x)dx+g(y)dy = 0 for every piecewise-smooth simple closed curve C. Consider the following function: f(x)=\left\{\begin{matrix} x^2sin\frac...
(There are ways to make the use of infinitesimals rigorous, such as non-standard analysis, but this is not the focus of my post today.) In single variable calculus, we learn that if we want to differentiate a function at some point x, then we need to compare the value f(x) of f ...
Curve Analyzed using calculus. Understanding the curve's slope at different points required differential calculus. 4 Arc Something shaped like a curve or arch The vivid arc of a rainbow. Curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line...
(calculus) The vector field denoting the rotationality of a given vector field. The curl of the vector field \vec{F}(x,y,z) is the vector field \operatorname{curl}\,\vec{F} \equiv \vec{\nabla}\times\vec{F}=\left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\...