Show that fi(r) = rª and f2(r) = r² +1 are orthogonal on the interval [–1, 1).

Q: Let f be a function such that Vf(x, y) = (2x, 1) g(x, y) = x² + y². Find all the points on the curve…

A: Let f be a function such that ∇f(x, y)=2x, 1 and let g(x, y)=x2+y2 We have to find all the points on…

Q: Let P denote the path that travels along the graph of y = x2 from (0,0) to (1,1). Let f(x,y) = √1 +…

A: Let P denote the path that travels along with the graph of y=x2 from (0, 0) to (1, 1). Let f(x,…

Q: 3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at…

A: Step 1: Step 2: Step 3: Step 4:

Q: Show that the into R² defined by T₁ = ( 9₁ ₂1₂) — 172,9₁) and T2: (F₁, F₂) → (Y¶₁₂ Y {₂) : are…

A:

Q: Let the contour : [0,3] → C be given by 7(t) = { 11 (1) = 2eint, 0<t≤1 Y2(t) = 2(t − 2), 1<t≤3. i.…

A:

Q: 5|, B= -14|, and C = |2 2 -11 B and C linearly dependent, or are they linearly independent? -arly…

A:

Q: (c) If f is a C² scalar function, then V × (Vƒ) = 0. True False (d) The lines 1₁ (t) = (1 − t, 1+ t,…

A: (C) If c is scalar function Means      curl(grad ƒ)  is = 0   Therefore correct ans is  True

Q: 1 -2 4 17. Let a 4 a2 -3 and b : For what 7 value(s) of h is b in the plane spanned by aj and a2?

A:

Q: Prove that the line x +1 = 글(-6) - lies in the plane 3x + y – 2 = 3.

A:

Q: ii. Use the basic definition of complex integration to evaluate the integral I $, z¹ dz, where the…

A: Step 1: Step 2: Step 3: Step 4:

Q: Verify that the given complex function is an Analytic Function, w = f(z) = 2z³ – 2z + 12

A:

Q: THEOREM 4.4 (LINEAR TRANSLATION) Let Y= aX + b; then: My(0) = eb" My(a0).

A:

Q: Find the plane through P = (4, −1, 9) containing the line r(t) =〈1, 4, −3〉+ t〈2, 1, 1〉.

A: The plane contains the line r(t)=1, 4, -3+ t2, 1, 1, so a vector in the plane is 2, 1, 1. The line…

Q: d s: X = (0,1, -1) + λ (1, α, 2α) are coplanar, in which case study its relative position.

A: Consider the given lines. r: X=(1,0,2)+λ (2, 1, 3) And, s:X=(0,1, -1)+λ(1, α, 2α) Let r and s are…

Q: Let L₂ be the line passing through the point P₁=(-3, -1, -2) with direction vector -[0, 3, 3], and…

A:

Q: Verify that the function f(z) = Z is continuous throughout the complex plane.

A:

Q: Compute ∇ƒ(-1, 2, 1), where ƒ(x, y, z) = xy/z.

A: Given,f(x,y,z)=xyz     As we know ∇f(x,y,z)=∂f∂x,∂f∂y,∂f∂z

Q: Let f(z) = x? + iy² where æ and y are the real and imaginary parts, respectively of z and C is a…

A:

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Let F = Vf and f = 5x²y - 5z. Calculate F. dr for the path r₁ = (t, t,0), 0 ≤ t ≤ 1. (Give your…

A: We will use fundamental theorem of line integrals

Q: Let c denote an arbitrary constant. Determine the orthogonal trajectory of x+xIn(y) =c that passes…

A:

Q: Consider Flx,y) = (3x2 +6y) i t (6x +4y%)j. (as Show that Fis lonservative a function Such Line…

A:

Q: Given the vector: j(x) sin(2x) X = X ,ex, 1-√1+x (2, 1,-2), x = 0 Determine its domain and show it…

A:

Q: Calculate the rectangular approximations R1, R2, and R4 for f(x)=2x−1 over the interval [1,5].

A:

Q: Verify that the function u(x.y) x- 3xy -5y is harmonic in the entire complex plane ,And find the…

A:

Q: Use the function to find the image of v and the preimage of w. T(Vg, V2) = (V, + V2 V1 – V2), v =…

A:

Q: Use the Cauchy-Riemann equations to show that the function f(z) - e² is not analytic anywhere.

A: The solution is given as

Q: Use the function to find the image of v and the preimage of w. T(V1, v2) = (Vị + v2, V1 - V2), v =…

A:

Q: Suppose v,, v,,V3 are distinct points on one line in R" . The line need not pass through 29 the…

A: