Continuity Definition and 909 Threads

How does equation of continuity relate to conservation of mass principle? Laymans terms please.

Hello! I was wondering if anyone could expand upon and help me with this as I'm struggling "Use continuity to evalute \lim_{{x}\to{\pi}}\cos(x+\sin(x))" I do remember faintly how to do limits of "normal" numbers, but with trig I did not learn at all so I'm confused. This is same as finding the...

Homework Statement Homework Equations lim(x,y)->(a,b)f(x,y) continuous at (a,b) if lim(x,y)->(a,b)f(x,y)=f(a,b) Squeeze theorem if lim a=lim c and lim a<= lim b <= lim c then lim b= lim c The Attempt at a Solution I proved that all the limits exist but somewhat the functions aren't all...

Homework Statement In a jet engine, a flow of air at 1000 K, 200 kPa, and 40 m/s enters a nozzle, where the air exits at 500 m/s and 90 kPa. What is the exit temperature, inlet area, and exit area, assuming no heat loss? Homework Equations min = mout = m where m = mass air flow dE/dt cv = Qcv...

Homework Statement Determine all values of the constant a such that the following function is continuous for all real numbers. f(x) = ax/tan(x), x ≥ 0 = a2 - 2, x < 0 Homework EquationsThe Attempt at a Solution I tried so many different ways to get the first part of the function to be...

Im uncertain if this belongs here, so I'm sorry in advance. I am extremely confused with continuity. I've tried going over my notes, reading the section, I've read about them in a separate book, I've used Khan Academy, I've talked to my professor, and I've gone to a tutor. But, to no avail on...

Hi, I am studying the Metal-Semiconductor junction. I cannot find any clear explanation of the continuity of the vacuum level at the interface. Can someone help me? My reference book provides me with the following: When two different materials are brought into contact, they must share the...

Quoted from Wikipedia, A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X. How to comprehend this definition in a intuitive way?

This is picture taken from my textbook. I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous...

Homework Statement ##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##: The...

A pipe is discharging from H=100m to an open atmosphere. The available discharge at inlet is 0.4m3/s 1. The bernauli's equation tells v=sqrt(2*g*H), neglecting any losses in the pipe, v=~44m/s 2. continuity equation tells v=Q/A, say diameter of pipe=0.3m, v=~5.65m/s which is correct?

The continuity equations are ##\nabla_a T^{ab} = 0##. In a coordinate basis, we can write out the resulting differential equations as: ##\nabla_a T^{ab} = \partial_a T^{ab} + \Gamma^a{}_{ac}T^{cb} + \Gamma^b{}_{ac}T^{ac}## (modulo possible typos, though I tried to be careful-ish). What do we...

Hi, I'm trying to understand vortex shedding and how the Karman vortex street occurs when air flows around a cylindrical object, so far it's going OK but then I came across this part of the explanation which leaves me confused: "Looking at the figure above, the formation of the separation...

Can we apply Euler's Equation and continuity equation to compressed air flow from tank or orifice.

I would appreciate it if someone could explain the steps in the reasoning of the following statement. This is not a homework assignment or anything. Let ##U \subseteq R^{n}## be compact and ##f:U\to R## a continuous function on ##U##. However f is not uniformly continuous. Then there exists an...

I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be...

I claim that if a function ##f:\mathbb{R}\rightarrow\mathbb{R}## is continuous at a point ##a##, then there exists a ##\delta>0## and ##|h|<\frac{\delta}{2}## such that ##f## is also continuous in the ##h##-neighbourhood of ##a##. Please advice if my proof as follows is correct. Continuity at...

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.)----------Proposition: Let $A$ be a countable subset of the open interval $(a,b).$ Then there is an increasing function on...

When a hose with running water is partially blocked with our finger ,the water comes out with a greater velocity. This is in agreement with the law of continuity in fluids which states that velocity of fluid is inversely proportional to the area of cross section. But when a running tap is...

Suppose their is a bucket with water inside it and there is a small hole at the bottom of the bucket such that water leaks from the end. Area of cross section of the bucket is A and area of the small hole is a. The velocity with which water is coming out of the hole is v and the velocity with...

Hello, I've been attempting to do these problems from my textbook: 1. Suppose that f is a continuous function on a bounded set S. Prove that the following two conditions are equivalent: (a) The function f is uniformly continuous on S. (b) It is possible to extend f to a continuous function on...

Homework Statement a) Show that the function f(x,y)=\sqrt[3]{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...

Homework Statement Hello, thank you in advance for all help. This is a limit problem that is giving me a particularly hard time. Homework Equations For what values of a and b is f(x) continuous at every x? In other words, how to unify the three parts of a piecewise function so that there are...

Homework Statement I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)} The problem had three parts. The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2) The second was to prove that the function f(x) was continuous...

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

Homework Statement https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xap1/v/t1.0-9/10923273_407123639463753_2874228726948727052_n.jpg?oh=27c882da16071e65bbb420147333ec38&oe=558413E4&__gda__=1434978872_d03c8531060688181560956b68c96650 Is f continuous at (0,0)? What is the "maximum" region D...

Hello! *Let $f$ be a strictly increasing continuous function on a closed interval $[a, b]$, let $c = f(a), d = f(b)$, and let $g:[c, d] → [a, b]$ be its inverse. Then $g$ is a strictly increasing continuous function on $[c, d]$.* How can it be shown that $g$ is continuous at its endpoints $c$...

Hi, I am trying to find the exact solution of the Continuity Equation. Any Idea how can i start solving it, i need it for some calculation in Image Processing. $$\pd{C}{t}+\pd{UC}{x}+\pd{VC}{y}=0$$ Where $U$ and $V$ is velocity in $X$ and $Y$ direction. The initial condition is as...

Homework Statement Derive a mathematical relationship which encapsulates the principle of continuity in fluid flow. Homework EquationsThe Attempt at a Solution Imagine we have a mass of fluid ## M##, of volume ##V##, bounded by a surface ##S##. If we take a small element of this volume...

f(x) = x^3 - 12x^2 + 44x - 46 x from 1 to 7 The attempt at a solution: f(1) = -13 f(2) = 2 f(4) = 2 f(5) = -1 f(6) = 2 So naturally, the answer should be: (1,2) U (4,5) U (5,6) right? Well, it didn't accept this answer. I think there is something wrong with whatever that is accepting the...

I need help with the proof of Apostol Theorem 5.2. Theorem 5.2 and its proof read as follows: https://www.physicsforums.com/attachments/3910 In the above proof, Apostol gives an expression or formula for f^* and then states the following: " ... ... Then f^* is continuous at c ... ... "I need...

I am reading Tom Apostol's book: Mathematical Analysis (Second Edition). I am currently studying Chapter 4: Limits and Continuity. I am having trouble in fully understanding the proof of Bolzano's Theorem (Apostol Theorem 4.32). Bolzano's Theorem and its proof reads as follows...

I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is- The indefinite integral of a function x with respect to f(x) is another function g(x) whose derivative is f(x). i.e. g'(x) = f(x) ⇒ Indefinite...

Dear experts, While solving the wave transmission at an interface for an acoustic wave problem, a boundary condition states that the "velocity of a fluid particle at the surface must be continuous". Could you please let me know why is it required, and a physical insight of what would happen if...

Let $K \subset \mathbb{R^n}$ be compact and let $f: K \rightarrow \mathbb{R}$ be continuous. Suppose that $f(x) > 0$ $\forall x \in S.$ Prove there is a $c > 0$ such that $f(x) \geq c$ $\forall x \in K$ My Sol: I said that by the extreme value theorem $\exists a,b \in K $ such that $f(a)...

Hello everyone, I have a rather simple question. I have the curve ## C(t) = \begin{cases} 1 + it & \text{if}~ 0 \le t \le 2 \\ (t-1) + 2i & \text{if }~ 2 \le t \le 3 \end{cases} ## which is obviously formed from the two curves. This curve is regarded as an arc if the functions ##x(t)## and...

Hi If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e. f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2) and f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2) for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##, does it mean that it is continuous...

Homework Statement Use Maxwell's equations to derive the continuity equation. B=Magnetic Field E=Electric Field ρ=Charge Density J=Current Density Homework Equations Maxwell's Equations: ∇⋅E=ρ/ε0, ∇×E=-∂B/∂t ∇⋅B=0 ∇×B=ε0μ0(∂E/∂t)+μ0J Continuity Equation: ∇⋅J +∂ρ/∂t = 0 The Attempt at...

For the potential V(x)=V1(x)+iV2(x) the continuity equation yields: ∇⋅j=-∂ρ/∂t + 2*ρ*V2/ħ (unless I am mistaken). What is the interpretation of this result?

Hi! Just curious... Does a break in the continuity of a conductor's volume affect its magnetic field, resistance, etc?

Homework Statement Suppose the function ##f:[0,1] \to \mathbb{R}## is continuous, ##f(0) > 0## and ##f(1)=0##. Prove that there is a number ## x_0 \in (0,1] : f(x_0) = 0## and ##f(x) > 0## for ##0 \leq x \leq x_0##. Homework Equations We can't use the IVT. Additionally, the definition of...

Homework Statement Daphne goes to the kitchen for a glass of water. She turns on the faucet so the stream of water is laminar flow. She notes that the diameter of the water steam decreases with distance below the faucet assuming that the water exits the faucet of diameter D with speeds v0...

Hey there, I trying to understand the following coordinate transformation of the equation of continuity (spherical coordinates) for a vaporizing liquid droplet\frac{\partial \rho}{\partial t} + \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \rho v) = 0 into \epsilon \sigma \frac{\partial...

Based on the following problem from http://math.uchicago.edu/~vipul/teaching-0910/151/applyingformaldefinitionoflimit.pdf: f(x) = \begin{cases} x^2 &, \text{ if }x\text{ is rational} \\ x &, \text{ if } x\text{ is irrational} \end{cases} is shown to have the following limit: \lim_{x\to 1}f(x)...

Homework Statement A function f is defined as follows: ƒ(x) = sin(x) if x≤c ƒ(x) = ax+b if x>c Where a, b, c are constants. If b and c are given, find all values of a (if any exist) for which ƒ is continuous at the point x=c. Homework EquationsThe Attempt at a Solution I was unsure of how to...

Homework Statement If f(x) be a "continuous" function in interval [a,b] such that f(a)=b and f(b)=a, then prove that there exists at least one value "c" in interval (a,b) such that f(c)=c. Note: [a,b] denotes closed interval from a to b that is a and b inclusive. (a,b) denotes open interval...

im lost. can someone explain how to do this problem? determine the set of points at which the function is continuous

1/ Prove that the set-valued map F defined by F : [0, 2π] ⇒ R2 as F(α) := {λ(cos α, sin α) : λ ≥ 0}. is continuous, but not upper semicontinuous at any α ∈ [0, 2π]. 2/ What is the fact that " F is continuous if it is both u.s.c. and l.s.c". I would like illustrate that and thank you.

Homework Statement Examine the continuity of P: x -----> tan(Pi/x) 2. The attempt at a solution: I considered fog with f=tan(x) and g=Pi/x But the problem I have is with the domain of definition can someone please help !

The question looks like this. Let $f(x, y)$ = 0 if $y\leq 0$ or $y\geq x^4$, and $f(x, y)$ = 1 if $0 < y < x^4 $. (a) Show that $f$ is discontinuous at (0, 0) (b) Show that $f$ is discontinuous on two entire curves. In regarding (a), I know $f(x, y)$ is discontinuous on certain...