Multivariable calculus Definition and 276 Threads

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one.

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1. 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. 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. 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. 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. 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. 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. 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...

12. 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.

38. 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...
39. 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...
40. 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}##...
41. 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...
42. 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...
43. 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|}...
44. 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}...
45. 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?
46. 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...
47. 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...
48. 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...
49. 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...
50. Multivariable Calculus, plane sketching

How do I know where to put the axes for the equation 4x^2 - 9y^2 = z when graphing in 3d?