Vector calculus Definition and 422 Threads

Dear all, I'n an EE that finished his degree more than 10 years ago. I wanted to refresh my Electricity and Magnetism knowledge. I bough Purcells book some weeks ago (https://www.amazon.com/dp/1107014026/?tag=pfamazon01-20) and I'm kind of struggling through the maths (Vector calculus). I've...

Homework Statement Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point ##(x,y)## is given by ##M(x,y)=3x^2+y^2+5000##. If Igor is at the point ##(8,6)##, describe the curve along which he should travel if he wishes to reduce the field...

Suppose we have do a curl of two 2-d vectors... we get the 3rd axis about which it is rotating. But when we do the curl of two 3-d vectors.. we get a answer like x-y plane is rotating wrt z axis, y-z plane rotating wrt to x-axis and similarly x-z plane rotating wrt to y axis. My question is...

Homework Statement Let ##F(x,y)=4sin(xy)+x^3+y^3## Use Newton's method to approximate the critical point that lies near ##(x,y)=(-1,-1)## Homework EquationsThe Attempt at a Solution I have a problem here because the derivative is not a square matrix. Hence, I can't find the inverse needed for...

Homework Statement Let $$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2} \space & \text{if} \space(x,y)\neq(0,0)\\0 \space & \text{if} \space(x,y)=(0,0)\end{cases}$$ a) Use the definition of the partial derivative to find ##f_x(0,0)## and ##f_y(0,0)##. b) Let a be a nonzero constant and let...

Homework Statement Let ##g(x,y)=\sqrt[3]{xy}## a) Is ##g## continuous at ##(0,0)##? b) Calculate ##\frac {\partial g}{\partial x}## and ##\frac{\partial g}{\partial y}## when ##xy \neq 0## c) Show that ##g_{x}(0,0)## and ##g_{y}(0,0)## exist by supplying values for them. d) Are ##\frac...

Homework Statement Function is ##f(x,y)=((x-1)y)^\frac{2}{3}##,##\space\space(a,b)=(1,0)## a) Calculate ##f_{x}(a,b)## and ##f_{y}(a,b)## at point ##(a,b)## and write the equation for the plane. Homework EquationsThe Attempt at a Solution So...

I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre...

Homework Statement Suppose that you have the following information concerning a differentiable function ##f##: ##f(2,3)=12##, ##\space## ##f(1.98,3)=12.1##, ##\space## ##f(2,3.01)=12.2## a) Give an approximate equation for the plane tangent to the graph of ##f## at ##(2,3,12)##. b) Use the...

Homework Statement Let ##f(x,y) = \|x \| - \|y\| - |x| - |y|## and consider the surface defined by the graph of ##z=f(x,y)##. The partial derivative of ##f## at the origin is: ##f_{x}(0,0) = lim_{h \rightarrow 0} \frac{ f(0 + h, 0) - f(0,0)}{h} = lim_{h \rightarrow 0} \frac {\|h\| -|h|}{h} =...

Homework Statement Function is ##lim_{(x,y,z) \rightarrow (0,\sqrt\pi,1)} \ e^{xz} \cos y^2 - x## Homework EquationsThe Attempt at a Solution As ##x \rightarrow 0## along ##y= \sqrt \pi, z=1##, ##f(x,y,z)= -1## As ##y \rightarrow 0## along ##x=0, z=1##, ##f(x,y,z) = -1## As ##z \rightarrow...

Homework Statement This is the function: ##\lim_{(x,y) \rightarrow (0,0)} \frac{(x+y)^2}{x^2+y^2}## Homework EquationsThe Attempt at a Solution So for ##x \rightarrow 0## along ##y=0##, ##f(x,y)=1## For ##y \rightarrow 0## along ##x=0##, ##f(x,y)=1## also. But the answer says there is no...

Homework Statement Examine the behavior of ##f (x,y)= \frac{x^4y^4}{(x^2 + y^4)^3}## as (x,y) approaches (0,0) along various straight lines. From your observations, what might you conjecture ##\lim_{(x,y) \rightarrow (0,0)} f(x,y)## to be? Next, consider what happens when ##(x,y)## approaches...

Homework Statement Equation of ellipsoid is: ##\frac{x^2}{4} + \frac{y^2}{9} + z^2 = 1## First part of the question, they asked to graph the equation. I have a question about this, I know that ##-1\leq z \leq 1##. So what happens when the constant 1 gets smaller after minusing some value of...

Homework Statement a) Suppose ##g## is a function such that the expression for ##g (x,y,z)## involves only ##x## and ##y## (i.e., ##g (x,y,z)=h (x,y)##). What can you say about the level surfaces of ##g##? b) Suppose ##g## is a function such that the expression for ##g (x,y,z)## involves...

Homework Statement Suppose that a surface has an equation in cylindrical coordinates of the form ##z=f(r)##. Explain why it must be a surface of revolution. Homework EquationsThe Attempt at a Solution I consider ##z=f(r)## in terms of spherical coordinates. ## p cosφ = f \sqrt{(p sinφcosθ)^2...

Homework Statement The surface is described by the equation ## (r-2)^2 + z^2 = 1 ## in cylindrical coordinates. Assume ## r ≥ 0 ##. a) Sketch the intersection of this surface with the half plane ## θ= π/2 ## Homework Equations ## r= psin φ ## ## p^2 = r^2 + z^2 ## The Attempt at a Solution...

Homework Statement Show that for any scalar field α and vector field B: ∇ x (αB) = ∇α x B + α∇ x BHomework Equations (∇ x B)i = εijk vk,j (∇α)i = αi (u x v)i = eijkujvk The Attempt at a Solution Since α is a scalar i wasn't quite sure how to cross it with ∇ So on the left side I have...

So I have a quick question that will hopefully yield some clarification. So the divergence of a dyadic ##\bf{AB}## can be written as, $$\nabla \cdot (\textbf{AB}) = (\nabla \cdot \textbf{A}) \textbf{B} + \textbf{A} \cdot (\nabla \textbf{B})$$ where ##\textbf{A} = [a_1, a_2, a_3]## and...

The magnetic field generated by an infinitely long straight wire represented by the straight line ##\gamma## having direction ##\mathbf{k}## and passing through the point ##\boldsymbol{x}_0##, carrying a current having intensity ##I##, if am not wrong is, for any point ##\boldsymbol{x}\notin...

Homework Statement Hey I'm trying to prove the rigorous definition of limit for the following function: Lim (x,y) approaches (1,1) of f(x,y)=(y*(x-1)^(4/3))/((x/1)^2+abs(x)*y^2) Homework Equations abs(x^2)<abs(x^2 +y^2) The Attempt at a Solution I know the rigorous definition of limit. I...

Hello, do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)? This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology). Thanks in advance!

Homework Statement How to I explain that maxwell's equation has well defined divergence Homework Equations All four EM Maxwell's equation The Attempt at a Solution I discussed it by showing one of the property of Maxwell's equation that is the Divergence of a Gradient is always zero (With...

This is more of an intuitive question than anything else: the curl of a vector field \mathbf{F} , \nabla \times \mathbf{F} is defined by (\nabla \times \mathbf{F})\cdot \mathbf{\hat{n}} = \lim_{a \to 0} \frac{\int_{C} \mathbf{F}\cdot d\mathbf{s}}{a} Where the integral is taken around a...

Hi, I want to re-learn multivariable calculus, after I have learned it, not in the best possible way... and feel bad about it. I have seen the recommendations here about Hubbard/Shifrin/Fleming/Edwards. I have also seen the books by Munkres/Spivak/Apostol. I didn't really like Hubbard's book...

The following identity is found in a book on Turbulence: Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki. Thanks

Homework Statement Use index and comma notation to show: \begin{equation*}\text{div }(\text{curl } \underline{\bf{v}}) = 0\end{equation*} Homework Equations \begin{align*} & \text{(1) div } \underline{\bf{v}} = v_{i,i} \\ & \text{(2) curl } \underline{\bf{v}} = \epsilon_{ijk} v_{j,i}...

Homework Statement Evaluate ##\int\int_S \textbf{F}\cdot\textbf{n} dS ## where ##\textbf{F}=(z^2-x)\textbf{i}-xy\textbf{j}+3z\textbf{k}## and S is the surface region bounded by ##z = 4-y^2, x=0, x=3## and the x-y plane with ##\textbf{n}## directed outward to S. The attempt at a solution I've...

This is part of a larger question, but this is the part I am having difficulty with. I have had an attempt, but am not sure where I am making a mistake. Any help would be very, very appreciated. 1. Homework Statement Let C2 be the part of an ellipse with centre at (4,0), horizontal semi-axis...

Homework Statement We have a fluid with density ρ which is rotating about the z-axis with angular velocity ω. Where should a unit square, call it S, be placed in the yz-plane such that there is zero net amount of fluid flowing through it? Homework Equations...

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law essentially uses the fact that, if ##\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3##, ##\boldsymbol{J}\in C_c^2(\mathbb{R}^3)##, is a compactly supported twice continuously differentiable field...

In the vector calculus course, I calculated integrals like, ##\int \vec F \times \vec{dr} ## Does this kind of integrals have physical significance or practical application other than Biot-Savart's Law?

Could someone explain to me in simplest of terms what scale factor is when dealing with orthogonal vectors.

I'm not sure if the title correctly says what I am looking for. I'm a few years out of college and I'm trying to review some electromagnetics topics. A lot of the "proofs" in my EM book seem to take a lot of shortcuts, or use "intuition" to explain why some calculus operation can be simplified...

Hi, I am currently in the first year of my undergraduate mathematics degree and I am taking a course in vector calculus. The course content is: line integrals, conservative field, divergence, gradient, curl, the divergence theorem, green’s formula, Stokes' them., field theory. I have seen that...

Homework Statement Prove that a current density J(r, t) such that ∇ × J = 0 implies the magnetic field B = 0.Homework Equations Maxwell's equations, vector calculus The Attempt at a Solution I've played around with Maxwell's equations and with the properties of vector calculus but I can't...

My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are: \mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...

Hi, friends! I have been able to understand, thanks to Hawkeye18, whom I thank again, that, if ##\mathbf{J}## is measurable according to the usual ##\mathbb{R}^3## Lebesgue measure ##\mu_{\mathbf{l}}## and bounded, a reasonable hypothesis if we consider it the density of current, if...

Homework Statement Write in Vector-Matrix form then write the augmented matrix of the system. r + 2s + t = 1 r - 3s +3t = 1 4s - 5t = 3 Homework Equations The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the...

Homework Statement prove grad(a.grad(r^-1))= -curl(a cross grad (r^-1)) Homework Equations curl(a x b)= (b dot grad)a - (a dot grad)b +a(div b) - b(div a ) The Attempt at a Solution Im trying to use index notation and get di (aj (grad(r^-1))j) =grad(r^-1) di(aj) +aj(di grad(r^-1))j which is...

Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...

I'm learning vector calculus and am wondering how general it is. The appear to be using a smoothness condition, but what is it? Certainly the functions are required to have two derivatives. That is, the partial derivatives can be taken twice. Are they further required to have an infinite...

Got it. Thank you ;)

Homework Statement Verify the divergence theorem for the function V = xy i − y^2 j + z k and the surface enclosed by the three parts (i) z = 0, s < 1, s^2 = x^2 + y^2, (ii) s = 1, 0 ≤ z ≤ 1 and (iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1. Homework Equations [/B]...

I am currently studying a course on waves, which has a real ambiguity in the lecture notes. Essentially, I don't know how the professor got from equation \ref{eq:surf_x-y} to equations \ref{eq:vel_u} and \ref{eq:vel_w}. I have tried to work backwards to find a method but am not sure of its...

Homework Statement http://faculty.fiu.edu/~maxwello/phz3113/probs/set1.pdf I'm working on problem 2. Trying to prove that the dot product between a vector field and its curl is zero. Homework Equations The basic identities of vector calculus and how scalar fields and vector fields interact...

Why phi component is not taken into account while calculation electric field intensity due to line charge?See attachment for details.https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/74/74211-22d26b27211fd4b186955b890458804e.jpg...

Homework Statement 2. The shape of a hill is described by the height function h(x,y) = (2 + x2 + y4)-1/2 (a) Find the gradient ∇h(x,y) (b) What is the maximum slope of the hill at the point r0 = i+j [or (x,y) = (1,1)]? (c) If you walk north-east (in the direction of the vector i+j) from...

I've put the problem statement below and worked it out. I typically don't post questions like this as they're a lot to go through, but I am wondering if I have worked the problem correctly as my book does not have the solution and I feel like I am not understand the material correctly. 1...

Homework Statement I need to show that $$\del*\vec{A(\vec{r})}=\frac{\mu}{4\pi}\int{\frac{\vec{J{vec\r'}}}{\vec{R}}}d\tau=0$$ where A is the vector potential and R refers to "script r" or (r-r') where r is source point of charge and r' is the measurement point. tau refers to a volume integral...