Multivariable calculus Definition and 277 Threads

  1. P

    I Doubt about theorem in Calculus on Manifolds

    Here ##Df(a)## is the derivative of ##f## at ##a##, i.e. the linear transformation at . My question is simply; if the assumptions in the theorem hold, is the map ##a\mapsto Df(a)## also continuous? Spivak seems to only prove the existence, not the continuity. If it is true that ##a\mapsto...
  2. P

    Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds

    Proof: Suppose that ##\lim _{x \rightarrow a }f^i(x)=b^i## for each i. Let ##\epsilon>0##. Choose for each ##i##, a positive ##\delta_i## such that for every ##x \in A\setminus\{a\}## with ##|x-a|<\delta_i##, one has ##\left|f^i(x)-b^i\right|<\epsilon / \sqrt{n}##. Let ##\delta=\min...
  3. P

    I On inverse function theorem in Spivak's CoM

    I know of a thread on this site with a similar question, but no definite answer. I will not state the whole proof, as it is quite long. 1. Why can we assume ##f## to have the identity map as its derivative? I understand how if the theorem is true for ##g = \lambda^{-1} \circ f##, then its true...
  4. KungPeng Zhou

    Multivariable calculus proof involving the partial derivatives of an expression

    For the first equation: ##f(tx, ty, tz)=f(u, v, w) ##, ##u=tx, v=ty, w=tz##,##k=f(u, v, w) #### t^{n}f_{x}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}## As the same calculation ##xf_{x}+yf_{y}+zf_{z}=[\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}...
  5. S

    I Multivariable fundamental calculus theorem in Wald

    i want to prove that if ##F:\mathbb{R}^n\to\mathbb{R}## is a differentiable function, then $$F(x)=F(a)+\sum_{i=1}^n(x^i-a^i)H_i(x)$$ where ##H_i(a)=\frac{\partial F}{\partial x^i}\bigg|_{x=a}##. the hint is that with the 1-dimensional case, convert the integral into one with limits from ##0## to...
  6. S

    Question about arc length and the condition dx/dt > 0

    This is not homework That passage is from James Stewart (Multivariable Calculus). I want to ask about the condition dx/dt > 0. If dx / dt < 0, the formula can't be used? Thanks
  7. V9999

    I A doubt about the multiplicity of polynomials in two variables

    Let ##P(x,y)## be a multivariable polynomial equation given by $$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$ which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say, $$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$ My doubts relie on the multiplicity of this point...
  8. cwill53

    I Gradient With Respect to a Set of Coordinates

    In physics there is a notation ##\nabla_i U## to refer to the gradient of the scalar function ##U## with respect to the coordinates of the ##i##-th particle, or whatever the case may be. A question asks me to prove that $$\nabla_1U(\mathbf{r}_1- \mathbf{r}_2 )=-\nabla_2U(\mathbf{r}_1-...
  9. T

    Calculus Multivariable calculus PDF books

    Multivariable calculus is a branch of mathematics that extends the concepts of single-variable calculus to functions of multiple variables. In this subject, vectors and partial derivatives are introduced to represent and manipulate multi-dimensional data. The gradient of a function represents...
  10. R

    I Multivariable function optimization inconsistency

    Mentor note: For LaTeX here at this site, don't use single $ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane. I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
  11. luqman

    Coordinate Transformation (multivariable calculus)

    My Progress: I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change: $$ \frac{\partial}{\partial\omega} f =...
  12. H

    Calculating total derivative of multivariable function

    This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.
  13. L

    I Second derivative of chained function

    Let's say we have a function ##M(f(x))## where ##M: \mathbb{R}^2 \to \mathbb{R}^2## is a multivariable linear function, and ##f: \mathbb{R} \to \mathbb{R}^2## is a single variable function. Now I'm getting confused with evaluating the following second derivative of the function: $$ [M(f(x))]''...
  14. F

    "Trick" for a specific potential function defined with an integral

    Hello, To first clarify what I want to know : I read the answer proposed from the solution manual and I understand it. What I want to understand is how they came up with the solution, and if there is a way to get better at this. I have to show that, given a vector field ##F## such that ## F ...
  15. Poetria

    Chain rule (multivariable calculus)

    ##f_x=3*x^2+y## ##f_y=2*y+x## ##(3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1}## Well, I am not sure how to evaluate it. I got a wrong result by multiplying by 0.1, i.e. ##((3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1})*0.1## I guess it is trivial but I am lost. :(
  16. T

    I Vector and Plane Relationship in 3D

    I have a quick question. If a Vector is contained inside a plane, would the normal of the plane be orthogonal to said vector? Thank you!
  17. Poetria

    Good approximation - multivariable calculus

    I tried to use a Taylor series expanded at 3 and set to 3.01: https://www.wolframalpha.com/input/?i=27+++9+(-3+++x)^2+++(-3+++x)^3+++3+y^2+++y^3+++(-3+++x)+(27+++y^2)=3.01 I got the vector ## (\Delta x, \Delta y)= (0.37887, -0.54038)## It does give a desired result but it is marked as wrong...
  18. Poetria

    Partial differential (multivariable calculus)

    Intersecting the graph of the surface z=f(x,y) with the yz -plane. This is the option I have chosen, but it's wrong. I don't understand why. x is fixed so I thought the coordinates: y and z are left. I thought this source may be helpful...
  19. B

    Multivariable Calculus proof for Optics

    Part A) For part A I forgo breaking down the identity into it's component x, y, and z parts, and just take the r derivative treating r' as a constant vector. This seems to give the right answer, but to be entirely honest I'm not sure how I'd go about doing this component by component. I figure...
  20. Adgorn

    Calculus Looking for a rigorous multivariable calculus book

    Hello everyone. I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution...
  21. S

    Understanding Griffith's Velocity Argument for Charge Integration

    In Griffith’s section 10.3.1, when proving why there is an extra factor in integrating over the charge density when it depends on the retarded time, he makes the argument that there can only ever be one point along the trajectory of the particle that “communicates” with the field point. Because...
  22. R

    B Is Multivariable Calculus as Fun as Single-Variable Calculus?

    I've been studying calculus A and B on and off over the last ten years, and I'm starting to learn calculus again for fun as soon as I can get my hands on a textbook. I was wondering if multivariable calc is as fun as A and B have been so far.
  23. cwill53

    Integrating Mass of a Hollow Sphere: Multivariable Calculus Explained

    I know some multivariable calculus, I just want someone to walk me through the integration deriving the mass element dM and the integration of thin rings composing the hollow sphere. It would also be nice if you could show me doing it one way using the solid angle and one way without using the...
  24. WonderKitten

    I Multivariable limits - path problem

    Hey, so I have the following problem: I'm trying to prove that the limit doesn't exist (although I'm not sure if it does or not) so: along y=mx -> x=y/m: , which is 0 for all k≠0. along y^n it's the same and I'm not sure what I should do next. Could I set x = sin(y)? If I can, then the limit...
  25. Hamiltonian

    B Basic doubts in vector and multi variable calculus

    If say we have a scalar function ##T(x,y,z)## (say the temperature in a room). then the rate at which T changes in a particular direction is given by the above equation) say You move in the ##Y##direction then ##T## does not change in the ##x## and ##z## directions hence ##dT = \frac{\partial...
  26. Leo Liu

    Calculus What book should I get for multivariable calculus after Stewart?

    Hi. I just finished the single variable part of Stewart's calculus book which helped me to master AP calculus. Now I am planning to move on to non-rigorous multivariable calculus. However, I have found reading his book a bit painful since the book mainly focuses on problem-solving techniques...
  27. R

    How to solve the integral which has limits from (1,2) to (2,4)

    I have a question like this; I selected lambda as 4 (I actually don't know what it must be) and try to make clear to myself like these limits (1,2) and (2,4) is x and y locations I think :) If I find an answer for part one of the integral following, I would apply this on another: My...
  28. T

    Multivariable Calculus, Line Integral

    The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$ And the line integral $$ \int_{C} F \cdot dr $$C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty $$ How do I calculate this? Anyone got a tip/hint? many thanks
  29. O

    How to prove this statement about the derivative of a function

    My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
  30. Mina Farag

    Applying the implicit function theorem to a system of equations

    My attempt: According to the implicit function theorem as long as the determinant of the jacobian given by ∂(F,G)/∂(y,z) is not equal to 0, the parametrization is possible. ∂(F,G)/∂(y,z)=4yzMeaning that all points where z and y are not equal to 0 are possible parametrizations. My friend's...
  31. S

    Determine the range of a function using parameter differentiation

    The strategy here would probably be to find a differential equation that ##f## satisfies, but differentiating with respect to ##x## using Leibniz rule yields ##f'=\int_x^{2x} (-te^{-t^2x}) \ dt + \frac{2e^{-4x^3}-e^{-x^3}}{x}## Continuing to differentiate will yield the integral term again...
  32. S

    Prove that this mapping is a bijection

    How would one tackle this using the definition? (i.e. for some function ff that f(x)=f(y)⟹x=yf(x)=f(y)⟹x=y implies an injection and y=f(x)y=f(x) for all yy in the codomain of ff for a surjection, provided such x∈Dx∈D exist.) One can solve the system of equations for x1x1 and x2x2 and that...
  33. Beelzedad

    I Multiple integral Jacobian confusion

    Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. ___________________________________________________________________________ Consider the following multiple integral: ##\displaystyle B= \iint_S \Biggl( \iiint_{V'}...
  34. S

    Finding the limit of a multivariable function

    If one approaches the origin from where ##x_2=0##, the terms ##x^2_1x_2+x^2_2x_3## in the denominator equal ##0##. Substituting ##|\textbf{x}|^2## for ##t## yields the expression ##\frac{e^t-1}{t}##, which has limit 1 as ##\textbf{x}\to\textbf{0}## and thus ##t\to0##. So the limit should be 1 if...
  35. Beelzedad

    I Is my interpretation of this three dimensional improper integral correct?

    In Physics/Electrostatics textbook, I am in a situation where we have to find the electric field at a point inside the volume charge distribution. In Cartesian coordinates, we can't do it the usual way because of the integrand singularity. So we use the three dimensional improper integral...
  36. Adesh

    How to find the volume under a surface?

    I want to know that how can z=$$ \sqrt{1-x^2}$$ ever represent a surface? It graphs a curve in the x-z plane and the triangle lies in x-y plane so how can they contain a volume, they are orthogonal to each other. I have attached awn image which is drawn GeoGebra for the function...
  37. M

    I What if the Jacobian doesn't exist at finite points in domain of integral?

    Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...
  38. E

    I Showing that a multivariable limit does not exist

    I want to show that the limit of the following exists or does not exist: When going along the path x=0 the limit will tend to 0 thus if the limit exists it will be approaching the value 0 when going along the path y=0, we get an equation with divisibility by zero. Since this is not possible...
  39. M

    How shall we show that this limit exists?

    Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...
  40. M

    I Why Does the Electric Field Calculation Diverge Inside the Volume?

    Let: ##\nabla## denote dell operator with respect to field coordinate (origin) ##\nabla'## denote dell operator with respect to source coordinates The electric field at origin due to an electric dipole distribution in volume ##V## having boundary ##S## is: \begin{align} \int_V...
  41. D

    How to prove that ##f(x,y)## is not integrable over a square?

    I'm confused with how Riemann sums work on double integrals. I know that ##L=\sum_{i,j}fm_{ij}A_{ij}## and ##U=\sum_{i,j}fM_{ij}A_{ij}## where ##m_{ij}## is the greatest lower bound and ##M_{ij}## is the least uper bound and ##A_{ij}## is the area of each partition. ##A_{ij}=\frac{1}{n^2}## for...
  42. M

    I Why is this volume/surface integration unaffected by a singularity?

    ##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}## Now by Gauss divergence theorem: \begin{align} \iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...
  43. hnnhcmmngs

    But, as I said, you don't actually need the coordinates at all.

    Homework Statement Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that |u|=√2, |v|=√3, u is perpendicular to v, w=u×v. Homework Equations |w|=|u×v|=|u|*|v|*sinΘ The Attempt at a Solution [/B] Θ=90° |w|=(√2)*(√3)*sin(90°)=√(6) Then I tried to use u={√2,0,0}...
  44. hnnhcmmngs

    Vectors and scalar projections

    Homework Statement Let a and b be non-zero vectors in space. Determine comp a (a × b). Homework Equations comp a (b) = (a ⋅ b)/|a| The Attempt at a Solution [/B] comp a (a × b) = a ⋅ (a × b)/|a| = (a × a) ⋅b/|a| = 0 ⋅ b/|a| = 0 Is this the answer? Or is there more to it?
  45. J

    B Geodesic dome parametric formula

    I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found http://teachers.yale.edu/curriculum/viewer/new_haven_06.04.05_u#f, but unfortunately, it comes short of providing me the most needed information, and so far I...
  46. A

    Calculus Multivariable calculus without forms or manifolds

    Hi there all, I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...
  47. J

    B How do you create a + and π sign using multivariable (x,y,z)

    I am taking a high school multivariable calculus class and we have an end-of-semester project where we trace out some letters etc., except that they all have to be connected, continuous and differentiable everywhere. My group's chosen to do Euler's formula, but so far we are having problems...
  48. komarxian

    Multivariable calculus problem

    Homework Statement Find the points on the surface xy^2z^3=2 that are closest to the origin Homework EquationsThe Attempt at a Solution x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to...
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