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).

Multivariable functions are functions like $f(x,y)=xy^2$, functions that have two or more inputs but still only one output.

There exist multivariable limits, where $x$ and $y$ approach a value instead of just $x$.

In multivariable calculus, when writing $\frac{\mathrm{d}}{\mathrm{d}x} \ f(x,y)$, one does not assume $y$ to be a constant, instead it is assumed that $y$ is a function of $x$.

To indicate that $y$ is a constant, one should use $\frac{\partial}{\partial x} f(x,y)$. These are called partial derivatives.

One also has integrals of multivariable functions. We say that we integrate over a surface in case of a two-variable function. We write this as $$\iint_S f(x,y)$$

If the surface is a rectangle $S: [a,b] \times [c,d]$, and the function is continuous on this rectangle, then Fubini's theorem says that $$\iint_S f(x,y) = \int^b_a\int^d_c f(x,y) \mathrm{d}y \mathrm{d}x$$

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Partial derivative in gradient descent for two variables

I've started taking an online machine learning class, and the first learning algorithm that we are going to be using is a form of linear regression using gradient descent. I don't have much of a background in high level math, but here is what I…
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Intuitive interpretation of the Laplacian Operator

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian Operator (a.k.a. divergence of gradient)?
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Why is gradient the direction of steepest ascent?

$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a vector. Reading this definition makes me consider…
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What is the Jacobian matrix?

What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?
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What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w / \partial r$ and $\partial w / \partial…
Chibueze Opata
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What is the solution to Nash's problem presented in "A Beautiful Mind"?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve (obviously, an exaggeration). Nonetheless, one…
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What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. [You might imagine something like $f(x)=x^2$.] View the graph of…
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References for multivariable calculus

Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it…
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Difference between gradient and Jacobian

Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is? The gradient is a vector with the partial derivatives, right?
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If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

Let $a$ and $b$ be positive numbers such that $a+b=1$. Prove that: $$a^{4b^2}+b^{4a^2}\leq1$$ I think this inequality is very interesting because the equality "occurs" for $a=b=\frac{1}{2}$ and also for $a\rightarrow0$ and $b\rightarrow1$. I tried…
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(Theoretical) Multivariable Calculus Textbooks

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope this is not considered offensive.) There are many…
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If a two variable smooth function has two global minima, will it necessarily have a third critical point?

Assume that $f:\mathbb{R}^2\to\mathbb{R}$ a $C^{\infty}$ function that has exactly two minimum global points. Is it true that $f$ has always another critical point? A standard visualization trick is to imagine a terrain of height $f(x,y)$ at the…
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How would you discover Stokes's theorem?

Let $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$, and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$. Stokes's theorem states that $$ \int_C f \cdot dr = \int_S (\nabla…
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Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus): the indefinite integral $\int f$ (also known as the…
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Limit $\frac{x^2y}{x^4+y^2}$ is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of solving explicitly in $\mathbb R^2 $, and it went as…
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