Let f(x+ y) = f(x).s) Vx, ye R.Prove that f is derivable on R iff it is derivable at one point of R.

Q: Let f(x, y)= Jxy +x+xcos y. (a) Find fxy.

A: Given:  Let fx, y=xy+x+x cos y To Find:  To determine fxy.

Q: The function f(z) =e² is: u(x.y)=e*cos(y) o u(x,y) =e*sin(y) ¤ contiuous in СО discontiuous in CO

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Q: Let f(z) = u(x, y)+iv(x,y) be entire such that u(x, y) 4v(x, y). Show that f is constant. %3D

A: Here we use CRE equation to prove this

Q: Show that the function u e(xy) + Vry V(y/x) provides the quation = 0. dy?

A: Again differentiate,

Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

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Q: Let f(x, y) = . Use the Linear Approximation at an appropriate point (a, b) to estimate…

A: Given,f(x,y) = x2y2+1To estimate,f(4.02,0.93) using the LinearApproximation at an appropriate point…

Q: Let f be a differentiable function of one variable, and let f(p), where p = (x² + y² + z²)'/2. Show…

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Q: Find fr and fy- pry S(27, y) = /" 2e" dt %3D f.(r, y) = | f„(x, y) = | %3D

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Q: Consider F(x,y)=(4x^3+4y)i+(4x+6y^2)j Show that F is conservative, find a function f such that F=…

A: Given F(x,y)=(4x^3+4y)i+(4x+6y^2)j F is conservative as shown below

Q: Let y = 2x2 + 7x +3. If Aæ 0.3 at r 4, use linear approximation to estimate Ay %3D Ay 2 instructor

A: Linear approximation

Q: Use the definition of differentiability to show that f(x, y) = y² – xy+5x is differentiable…

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Q: 1. Let F(x, y) = 1 0 if x + 2y ≥ 1 if x + 2y < 1 Does F defines the DF in the plane?

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Q: Let f : R² → R with f(x,y) = x²+ y. Prove that f is differentiable at the point (0, 0)

A: Given function is f(x,y)=x2+y Since we know that, differentiability of function of two variable at…

Q: Consider f(x, y) = x³y+ 2xy? Approximate f(2.1, 1.03) by using the tangent plane.

A: Given f(x, y) = x³y + 2xy² Find f(2.1, 1.03) using tangent plane.

Q: bounded by Y=x3, V=1x i rotare about X=1

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Q: find a potential function for F or determine that F is not conservative. F=(z,x,y)

A: F is conservative when curl F = 0 Given that f=(z, x, y)=zi+xj+yk

Q: Given f(x, y) = et" Y° – a°y°, show that fæy = fyx· 93 fyæ- %3D

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Q: Suppose f(r) z2 + sin z and F is an antiderivative of f that sausfies F(T) 0. Then F(2) = ( -1)

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Q: Use Stoke's Theorem to evaluate · F · dĩ where F = (xz, xy, 3xz) where C is the boundary of the…

A: Using the Stoke's theorem we have to evaluate  ∮F.→dr→ where F→=xz,xy,3xz=xzi+xyj+3xzk such that the…

Q: Define f: R² → R by sin(ry) f(x,y) := TY if ry # 0, f(t,0) = 1 = f(0, t) Vt e R. Prove that f is…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z =0 and find…

A: Cauchy-Riemann Equations: Let f=u+iv be a complex valued function , then the Cauchy-Riemann…

Q: Let f be a differentiable function of one variable, and let w = f(p), where p = (x² + y² + z?)!/2.…

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Q: Use integration by parts to evaluate lim Sest f(1)dt. (Let u = Z→∞J f(t) and dv = est dt.)

A: As per the question we are given a limit of an integral, and we have to use the integration by parts…

Q: Let F(x, y) = G , y 1 )- Calculate the work against F required to move an object from (1, 1) to (3,…

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Q: If f(x, y)is integrable function on the rectangle R = [a, b] x [c, d] where [a, b] on the x- axis…

A: In the question it is asked to show how the two double integrals are equal to each other.

Q: Suppose f(z) = r + sin z and F is an antiderivative of f that satisfies F(#) = 0. Then F(z) =

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Q: Consider the function g : Z x Z → Z × Z given by g(x, y) = (x+ 3y, 2x – y). Is g injective? Pr -…

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Q: 2. Let f(r) = v (2 - r)2. Use the local linear approximation of f (x) at ro = V0.99 (1.01)². 1 to…

A: Given Curve is f(x) =x2-x2  find linear approximation at x0=1  when x =0.99 value of f(x) at  x0=1…

Q: S F(2,v) For any function f(x, y) continuous on the ry-plane, f(r, y) dædy. O True False

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Q: c) Let f(r, y) = Vlry| Prove that f(r, y) is not differentiable at point (0, 0).

A: Given a function : f(x,y)=xy We need to show that f(x ,y) is not differentiable at (0 ,0). Let's…

Q: a²f If f(x,y) = e" sin (r + y), then prove that дхду дудх %3D

A: Hi! You have posted two non-related subparts. As nothing is specified, we are answering only the…

Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: || x dA, R is bounded by y = x, 2x + y = 6 and the y-axis. R

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Q: Suppose that F(x,y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that дх…

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Q: Determine the vactor feld of F. F(z, y) = yi - j

A: This is a problem related to vector field calculation. Based on the general concept we will provide…

Q: Let f(z)=ex^2 = U(x,y)+iV(x,y) Verify Ux=Vy

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Q: Let u be a differentiable function of x. Use the fact that ∣u∣ = √u2 to prove that

A: Given that - u = u2          ............(1)To prove - ddxu = u'uuProof - Difefrentiating equation…

Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

Q: (a) Let f(r, y) = x - 3ry + 18y, where 3r?y- y- 6x = 0. Prove that f(r, y) has extrema at the points…

A: Calculate partial derivative of f(x,y) with respect to x and y ,

Q: Show that the function u = p(xy) + VrY v(y/x) provides the uation x2 = 0. dy?

A: First find the partial derivative of the given function with  respect to x :

Q: Show that the function u = p(xy) + Vry v(y/x) provides the quation = 0. dy?

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Q: If y (x, t) is a function of x and t, prove that i) Lat =py (x, p)- y (x, 0) ii) L- = p°y (x, p) –…

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Q: Let f(r, y) 3ry? when (x, y) + (0,0) and f(0,0) = 0. g²+2y! At which points is f continuous?

A: The given function is: This function is continuous at a point if at that point limit exists and the…

Q: If f(z) = u + iv is entire and u(x, y) is bounded above, apply Liouville's Theorem to exp(f(z)) to…

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Q: 44. Let f(x, y) = g(u), where u = x² + y² and g(u) is differentiable. Prove that 2 4u (dg Әх du

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Q: Define f : [0, ∞) → [0, ∞) by f(x) = Vx. (a) (b) Prove that va +b 0. Prove that f is uniformly…

A: We have to prove that   √a+√b≤ √(a + b) for all a, b ≥ 0.

Q: If f(z) = sin(z 2) = U(x, y) + iV (x, y), determine U and V . Verify that the C-R conditions are met…

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Q: . Assume that f(x, y) is continuous on [a, b) x [c, d] and 9(r, v) = | 5(6,t)dt ds for a <a < b, c <…

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Q: Study the continuity of f(x,y) = xe-Vx at p(1,0) ey.

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Q: Let f(r, y) = x sin ry. Fi (a) fz(x,y) (b) fy(r, y) Sy(x, y) (c) fyy(", y) YY (d) fyr(r, y)

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.