Vector function Definition and 62 Threads

Homework Statement f:Rn->Rn is continuous and satisfies |f(x)-f(y)|>=k|x-y| for all x, y in Rn and some k>0. Show that F has a continuous inverse. Homework Equations The Attempt at a Solution It is easy to show that f is injective, but I've no idea how to prove the surjectivity. I...

Homework Statement suppose f is a differentiable mapping of R1 into R3 such that |f(t)|=1 for every t. Prove that f'(t)\cdot f(t)=0. I guess it is more proper to write (\nabla f)(t) \cdot f(t)=0, where (\nabla f)(t) is the gradient of f ant t. Homework Equations The Attempt at a...

missed a lecture and now have this homework problem and don't even know what the upside down triangle symbol indicates, can someone please give me a hand getting started, thanks consider the vector function q=(1/4X^4 y^2 z, x^3 yz^6 - cosh(xz), 1/7x^3 z^7) calculate f(x,y,z)=upside down...

Okay I know the definition of a Vector and of a scalar... but I am getting a little confused for some reason. Wolfram.com gives this definition of a scalar function: A function f(x_1,x_2,...,x_n) of one or more variables whose range is one-dimensional, as compared to a vector function...

Homework Statement If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t) show that the curve lies on a sphere with center at the origin Homework Equations The Attempt at a Solution I have no idea how to even approach this...

A vector function can be decomposed to form a curl free and divergence free parts: \vec{f}(\vec{r})=\vec{f_{\parallel}}(\vec{r'})+\vec{f_{\perp}}(\vec{r'}) where \vec{f_{\parallel}}(\vec{r'}) = - \vec{\nabla} \left( \frac{1}{4 \pi} \int d^3 r' \frac{\vec{\nabla'} \cdot...

Hello, I have a math problem that I think I've worked out properly, but I'm not entirely sure. The explanation is a bit lengthy, but I don't want to miss anything that might be pertinent. Essentially, I have a force equation F(t) that describes the acceleration of a body in two dimensions...

please help me, I try to do but i can not. 1. Find a vector function F(t) whose graph is the curve of intersection of z=\sqrt{4-x^2-y^2} and y=x^2. 2. Find parametric equations for the line that is tangent to the curve r(t)=(e^t)i+(sin t)j+\ln(1-t)k at t=0. 3. Find the point on the...

Homework Statement For each state wheather the function is a linear vector function of \vec{v} Homework Equations 1. \vec{F}(\vec{v})=\alpha \vec{v} 2. \vec{F}(\vec{v})= \vec{a} \times (\vec{b} \times \vec{v}) + (\vec{a} \times \vec{v}) \times \vec{v} The Attempt at a Solution I don't get...

Homework Statement how to find the domain of an vector function? Homework Equations The Attempt at a Solution

My related questions 1 Is there any difference between 'vector field' and 'vector function'? 'vector function' is also called 'vector-valued function' (Thomas calculus). According to their definitions, they are all the same things to me. And they are all some kind of mapping, which assigns a...

Original question: a) Find a vector function for the curve of intersection of the paraboloid z = 3x^2 + 2y^2 and the cylinder y = x^2. b) Show that this curve passes through (1,1,5) but not (3,3,9). I really have no idea how to do either parts of this question. Any help would be greatly...