Questions tagged [multivariable-calculus]
Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).
36,826 questions
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Boundary of points that satisfy constraint equations
I'm looking for a clean numerical approach to find the boundary of (x,y,z) points.
(happy to write this in latex if needed)
- 133
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Will Newton's method always be able to find every root of a multivariate system of equations?
Basically, if we have a system of equations $F(x)=0$ for some $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ that is globally twice continuously differentiable with finitely many roots, is there a guarantee ...
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What does Keisler mean by "touch" in his discussion about tangent planes?
I am reading a short section on p. 666 of the third edition of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler about tangent planes. He defines a tangent plane to a smooth function ...
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How to show that the vector of directional derivatives is tangent to a vector-valued function?
Let
$$
f : I_1 \times I_2 \times \dots \times I_n \to \mathbb{R}^m, \quad
f(x_1, \dots, x_n) = \big(f_1(x_1, \dots, x_n), \dots, f_m(x_1, \dots, x_n)\big),
$$
where each $f_i$ is continuous and ...
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The same triple integral over the same volume gives different results based on different limits that give the same volume.
I have this triple integral: $$I=\iiint_V xy\sqrt{z}$$
where $V$ is the volume constrained by the following surfaces:$$z=0,\,y=x^2, z=y,y=1$$
When I first tried to solve it , I understood the volume $...
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Is the Quotient of Two Coprime Polynomials in $\mathbb{R}[x,y]$ With Isolated Zeroes at the Origin Ever $C^\infty$?
Suppose $p(x,y)$ and $q(x,y)$ are two polynomials with real coefficients, that $p$ and $q$ have no common factor, and that $(0,0)$ is an isolated zero for both $p$ and $q$ in the sense that $p(x,y)$ ...
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Prove that the Euclidean norm is not differentiable at the origin
Let the scalar field $f : {\Bbb R}^2 \to {\Bbb R}$ be defined by $f(x,y) := \sqrt{x^2 + y^2}$. Prove that the partial derivatives of $f$ at $(0,0)$ do not exist and, thus, the gradient $\nabla f (0,0)$...
- 97
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Finding partial derivatives with direct and indirect dependencies
A student I am tutoring in multivariable calculus was given the following problem:
Suppose z = f(x, y), and we have an equation F(x, y, z) = k for some function F and
constant k. Use the chain rule to ...
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Average distance between all the points on 3d surface
I am trying to calculate the average distance a particle passing through a cylinder experiences. There is both a top and a bottom and the dimensions of the cylinder are known. Particles can exit any ...
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Why is curl a scalar in $\mathbb{R}^2$
In $\mathbb{R}^3$, for a vector field $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ $$\mathrm{curl}\ \mathbf{F}=\biggr(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\biggr)\mathbf{i}+\...
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Can I move the Lagrange multiplier to the other side?
Like instead of saying $$\nabla f=\lambda\nabla g$$ can I instead say $$\lambda\nabla f=\nabla g$$ to create an easier system of equations to solve?
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Is Riemann integrability of Banach-valued functions preserved under uniform limits?
Let $(f_n)$ be a sequence of Riemann integrable functions $f_n : [a, b] \to X$, where X is a Banach space. Here Riemann integrability is defined with the usual $\epsilon$-$\delta$ definition involving ...
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The surface $x + y + z = \sin xyz$: Can it be visualized?
The surface $$x + y + z = \sin xyz$$ is very hard to visualize:
What can be said about it, geometrically or visually?
Results so far
Start with a similar, but simpler, curve: $$x + y = \sin xy.$$ It ...
- 6,283
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Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
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Does each partial derivative vanish on the path from a local to a global minimum?
In $\mathbb{R}$, if there is only one critical point, and it is a local minimum, it is also the global minimum (because on every path to a different global minimum, there would have to be another ...
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