# Tagged Questions

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).

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### How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
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This is a two part question: 1) Let's define a recursive function as so: $$f(x,y)= \begin{cases} \hfill f(x,5) \hfill & y\le0 \\ \hfill 0 \hfill & y=1\\ \hfill x+f(x,y-1) \hfill & y>... 1answer 85 views ### Gradient of function of matrix exponential Suppose I have a differentiable function \phi: \mathbb{R}^{p\times p} \mapsto \mathbb{R} defined as \phi(\exp(tA)) where t is a positive scalar and A is a p\times p real matrix. How can I ... 1answer 33 views ### Can there be different values of y_p for one equation? For example, consider following example: Solution given by book is this: I solved it using different approach(as shown in the pic below) & got different answer. Is my solution wrong or ... 0answers 19 views ### Characterize |\nabla f| as minimal function which satisfies an upper gradient inequality Let f \in C^1( \mathbb R^n, \mathbb R) . Then one by chain rule has$$ (*)\qquad |f(g(1))-f(g(0))| \leq \int_0^1 |\nabla f|(g_t)|g'(t) |\ dt, \quad \forall g \in C^1([0,1],\mathbb R^n). $$I have ... 1answer 23 views ### Are scalar/vector fields in multivariable calculus related to fields of vector spaces in linear algebra In linear algebra, I have learned that vector spaces are defined over fields. I have to admit that I don't have any background in abstract algebra, so my knowledge of fields are limited to \mathbb R, ... 1answer 30 views ### Finding the maximum on an inside an octahedron Let B be the closed domain in \mathbb{R}^3 defined by |x_1|+|x_2|+|x_3|\leq 1. Find the maximum of F(x_1,x_2,x_3)=\sum_{i=1}^3x_i^2+\sum_{i=1}^3a_ix_i on B. Using Lagrange multiplier ... 1answer 36 views ### How to find stationary points of two-variable cubic [closed] I need to differentiate this cubic function to get the stationary points:$$f(x,y) = x^3 + ax^2 + bxy^2 + cxy + dx + e,$$where a, b, c, d and e are constants. How do I do this? 1answer 52 views ### Verification of a Diffeomorphism Below is an exercise to prepare for an Analysis II Exam Let f: \mathbb{R} \to \mathbb{R} be a function of Class C^1 such that |f'(t)| \leq k < 1 for all t \in \mathbb{R}. Show that the ... 2answers 69 views ### How to find extrema of \sqrt{x_1^2 + x^2_2 + x^2_3} defined on \{x \in \mathbb{R}^3 : x_1^2 + 2x^2_2 + 3x^2_3 < 1\} I have a function g: U \to\mathbb{R} where$$U :=\{x \in \mathbb{R}^3 : x_1^2 + 2x^2_2 + 3x^2_3 < 1\}$$and$$g(x) = \sqrt{x_1^2 + x^2_2 + x^2_3}$$I would like to find out if g(x) has any ... 0answers 22 views ### Finding the surface of intersection of 2 cylinders Let R=\{(x,y,z): x^2+z^2\leq 1, y^2+z^2\leq 1\}. Compute the area of its boundary \partial R. The formula is \int_D\sqrt{1+z_x^2+z_y^2}dxdy and z=\sqrt{1-x^2}, (I think). But what should the ... 3answers 91 views ### Prove an improper double integral is convergent I need to prove the following integral is convergent and find an upper bound$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$I've tried integrating \frac{1}{1+x^2+y^2} \lt \frac{1}{... 0answers 87 views ### Find the maximum volume of the pyramid bounded by the plane and the coordinate planes? Surface \sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z} , (c>0) I found that at (x_{0},y_{0},z_{0}) a tangent plane to the surface is : \frac{x-x_{0}}{2\sqrt{x_{0}}}+\frac{y-y_{0}}{2\sqrt{y_{0}}}+\... 2answers 42 views ### Let F be a vector field in \mathbb{R}^3. If F is divergence free, we may deform the surface. Why? In working through a solution, I can across the following generalization about vector fields and the Divergence Theorem. Can someone furnish a standard proof of this or at least its intuition? Let ... 2answers 211 views ### Maximize xy^2 on the ellipse x^2+4y^2=4 I was using Lagrange multiplier, any steps gone wrong?$$f(x,y)=xy^2c(x,y)=x^2+4y^2$$Partial Derivatives$$\frac {\partial f}{\partial x} = y^2 \frac {\partial f}{\partial y} = 2xy $$... 0answers 73 views ### Integral-Summation Problem (Mathematical Physics problem) given, ψ_{k} =\sqrt{\frac{2}{q}} \sin \frac{kπx}{q} we have, g_{kj}=q\int^q_0 ψ_j\frac{\partial ψ_k}{\partial q} dx verify, \sum_{k}g_{jk}g_{lk}=q^2\int^q_0 \frac{\... 0answers 23 views ### Approximation of a 2 variable function. If the weight of an object that does not float in water is x pounds in the air and its weight in water is y pounds , then the specific gravity of the object is : S= \dfrac{x}{x-y} For a certain ... 1answer 52 views ### Find the minimum value of (x+y) Two positive numbers x and y vary in such a way that \ 128x^2-16x^2y+1=0 Find the minimum value of (x+y). The answer is 35/4, how do I get the answer? 0answers 31 views ### Approximating a two variable function. A cylindrical tank is 4 feet high and has on outer diameter of 2 feet. The walls of the tank are 0.2 inches thick. We need to approximate the volume of the interior of the tank assuming that the tank ... 3answers 880 views ### Differentiability of a two variable function f(x,y)=\dfrac{1}{1+x-y} We're given the following function :$$f(x,y)=\dfrac{1}{1+x-y}$$Now , how to prove that the given function is differentiable at (0,0) ? I found out the partial derivatives as f_x(0,0)=(-1) and ... 2answers 53 views ### To find the critical points of f(x,y)=e^{-x}(x^{2}-5xy^{2}+4y^{4}) [closed] I am having hard time finding the critical points of$$f(x,y)=e^{-x}(x^{2}-5xy^{2}+4y^{4})$$, but i could not find. Can anyone help. Thanks EDIT When i substituted x=\frac{16}{10}y^2 in first ... 2answers 89 views ### Computing a double integral over a surface S, where S is the unit sphere,$$ \int \int_S (x^2+y^2)d\sigma$$Where S is the unit sphere centered at (0,0,0), and \sigma is surface area. I arrived at the correct answer of \large \frac{8\pi}{3}, but I took an (educated?) ... 1answer 84 views ### Stuck at the derivation of divergence in Cartesian coordinates. I'll get to the point immediately. The definition of divergence in a point (from my textbook):$$ div \bar{E} = \lim_{V \to 0} \frac{1}{V}\oint_S \bar{E}.d\bar{S}$$(it's a surface integral) ... 0answers 17 views ### What is the difference between ANOVA and ANOVA decomposition? I was reading the paper about ANOVA decomposition http://faculty.bscb.cornell.edu/~hooker/fame_jcgs.pdf but I can't see how it is related to ANOVA. (Except that we have mutual orthogonality between ... 2answers 59 views ### How to interpret \sum_{n\in \mathbb N^{d}} \frac{1}{n^{p}}; and when it is converges? I know that: \sum_{n\in \mathbb N} \frac{1 }{n^{p}} converges if p>1 and diverges if p\leq 1 My Question is: What is an analogue this in more than one variable (say d)? Does it make ... 3answers 286 views ### One Step Forward from Gaussian Integral Now to solve the integral  \int_0^\infty e^{-x^2} \, dx  has become a simple task for us. But how can we solve this integral:$$\int_0^\infty e^{-x^3} \, dx $$2answers 110 views ### Apparent discrepancy between change of variables in one versus multiple dimensions. My freshman calculus book gives the change of variables formula in one dimension and then eight chapters later gives it in n dimensions. But when it generalizes to n dimensions it requires the ... 3answers 168 views ### Hessian Matrix of an Angle in Terms of the Vertices I am attempting to derive the analytical formula for the Hessian matrix of a the second derivatives of the value of an angle in terms of the (9) coordinates of the 3 3D points that define it. While I ... 0answers 177 views ### Rudin's Rank theorem Rudin states the following: 9.32 Theorem: Suppose m,n, are nonnegative integers, m\geq r,n\geq r, F is a C^1 mapping of an open set E\subset R^n into R^m, and F'(x) has rank r for ... 0answers 38 views ### What happens to tangential gradient when flattening a surface The tangential gradient \nabla_\tau f associated to a surface S is defined as the projection of a suitable extension \nabla f to the tangent plane to that surface. It seems reasonable to think ... 1answer 27 views ### Tangent vectors and parametric curves Consider the curve C defined by (x,y,z) = \bar{r}(t), where$$\bar{r}(t)=\langle t\sin t, t\cos t, t^2 \rangle~~; t \in \mathbb{R}^3$$Show that C lies on the paraboloid z= x^2 + y^2 ... 1answer 110 views ### Find partial derivatives, given directional derivatives. [closed] You are given that the directional derivatives of a function f, at the point (a, b), in the direction of the two vectors (1, 2) and (−1, 1), are 2 and 3 respectively. Find the partial ... 4answers 293 views ### show that out of all triangles inscribed in a circle the one with maximum area is equilateral show that out of all triangles inscribed in a circle the one with maximum area is equilateral How do i start. I have to use function of two variables Thanks 1answer 55 views ### How do i determine maximum or minimum at (1,1) of function  f(x,y)=(x-y)^{4} + (y-1)^{4} How do i determine maximum or minimum at this point of function$$ f(x,y)=(x-y)^{4} + (y-1)^{4}$$I am getting doubtful case at point (1,1). How do i furthure investigate whether it is point of ... 0answers 55 views ### Open Set in the Cartesian plane. I'm trying to prove that the following set is an open set in \mathbb{R}^2:$$A=\{(x_{1},x_{2})\in\mathbb{R}^{2}: x_{1}+x_{2}>1\}$$with respect to norm ||x||_{1},||x||_{2},||x||_{\infty}. ... 2answers 28 views ### Use Implicit Differentiation to compute the partial derivate \frac{\partial z}{\partial x} at (1,1) The question I am working on is: The equation xy+z^3x-2yz=0 defines z as a function x,y around the point (1,1,1). Use Implicit Differentiation to compute the partial derivate \frac{\partial z}{\... 2answers 163 views ### Prove \{(x,y): x>0\} is connected As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that \{(x,y): x>0\} is connected. ... 1answer 39 views ### Is there a text introducing “high order Fréchet derivative” well? Let X,Y be Banach spaces and U be open in X. High-order Fréchet derivatives are defined inductively so that the n-th Fréchet-derivative of a function F is F^{(n)}:U\rightarrow L(X,L(X,....,L(... 2answers 116 views ### Condition for equality of mixed derivatives It says that theorems 12.11 and 12.12 imply Theorem 12.13. But, don't we need some extra conditions? Like existence of D_{r,r}f and D_{k,k}f? Here f is a function from \mathbb{R}^n to \mathbb{... 2answers 36 views ### Finding a good variable substitution for a double intergral I want to compute the following integral$$ \iint_D{(x-y) dxdy} $$where D is the triangle contained within these points: (0,0), (-2,1) and (-1,3). The lines that connect the the points form the ... 1answer 38 views ### Prove \lim_{(x,y)\to (1,-1)} f(x,y)=1 Let f : \mathbb R^2 \to \mathbb R be defined by f(x,y)= 3x + 2y. Prove that$$\lim_{(x,y)\to (1,-1)} f(x,y)=1$$I know that I must prove that for every \epsilon>0 we can find a \... 0answers 175 views ### Sufficient conditions for integration by parts in higher dimensions If \Omega\subset {\mathbb R}^n is a bounded open set with C^1 boundary and \nu denotes the outward unit normal to \partial \Omega, then the following formula holds for every pair of C^1 ... 2answers 34 views ### Proof of \frac{\partial f}{\partial\vec u}(x_0,y_0) = ∇f(x_0,y_0)\cdot \vec u  In order to prove that:$$\frac{\partial f}{\partial\vec u}(x_0,y_0) = ∇ f(x_0,y_0)\cdot \vec u $$my book defines:$$g(t) = f(x_0+at, y_0+bt)$$then, by the chain rule:$$g'(0) = \frac{\partial ...
In calculus, we have the following equation $DF(x,y)=\partial F_xdx+\partial F_ydy$ if $F$ is differentiable. I think such equation still holds for frechet derivative, but not for gateaux derivative. ...