Evaluate [ F. dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. Sc C: smooth curve from (-3, 2) to (3, 2) [4(2x + 3y)i + 6(2x + 3y)j] - dr .

Q: Find div F and curl F if F(x, y, z) = x²i – 5j + yzk. div F= curl F=

A:

Q: (a) Write the equation in slope-intercept form, and determine the slope and y-intercept. (b) Graph…

A:

Q: Derive the De Moivre's Theorem with detail Explanation?

A: Statement: For any real number x, we have (cos x + i sin x)n= cos(nx)  i sin(nx) Or, (eiθ)n=eiθn…

Q: Find L{f (t)}using the First Shifting Theorem, Note: a, b, e, w, k and n are constants 1. Il…

A:

Q: Find the two roots of (j)'/2 in rectangular. a) 0.770 + j0.770, –0.770 – j0.770 c) 0.707 + j0.707, –…

A: c

Q: #12- Fnd the indefinite integral of the follaung funchon and check the result by differentionion.

A:

Q: – x²y? x2 + y? f(x, y)

A: fx, y=-x2y2x2+y2 Now, z0=fa, b=f1, -2=-12-2212+-22=-45 Now partially differentiating fx,…

Q: c) Find the second degree polynomial approximation to f at (a, b) = (1, – 2) T2(x, y)

A: Given :- The second degree polynomial approximation to f at a,b = 1,-2 f x,y = -x2y2x2 +y2

Q: I of an analytic f hen the real part

A:

Q: se the given transformation to evaluate the integral. J 6z + 3ydA, where Ris the parallelogram with…

A: we have given: x=13u+v and y=132u-v  u=x+y, v=2x-y Now, Consider the integrals, ∬R6x+3ydA where, R…

Q: INSTRUCTION: Prove or disprove 1. Show that f: R-{-}→ R-{} defined by f(x) 3x-5 = is bijective. 2x+1

A: Given :  f: ℝ--12→ℝ-32   defined by ;                f(x) = 3x-52x+1 To prove : f is bijective.

Q: =x-y+3|

A: I have provided a graph of this in step2

Q: PewHite F in sphethal cordmates and compute its diergence.

A: The divergence of F is 3

Q: Let F (8xy, 3y, 8z). = The curl of F = (000). Is there a function f such that F = V f? ☐ (y/ (y/n)

A: Step 1: Step 2: Step 3: Step 4:

Q: where r is the position vector 6. Show that A) - da = | (Ax (Vf)) · da + fA- dl 7. Evaluate the…

A:

Q: For the vector field G = (yety + 4 cos(4x + y))i + (xey + cos(4x + y)) 5 I find the line integral of…

A:

Q: prove that ਰ Apply the calculus of residues to dx (1 + x²)² - π 4 .

A:

Q: HW1 2) Find the curl of F. VxF=

A:

Q: If F = Pi+Qj+ Rk, prove that div (curl F) = 0

A:

Q: Integral Representation r(c) Г(В)Г(c-b) Jo -2 F₁ (a, b; c; x) = for F, : ¹ (1 - ₁)¤-b-¹ (1-x₁)¯ª di…

A:

Q: Let ƒ(z) = |z|², z € C. Show that f is complex-differentiable at z = 0, but nowhere else.

A:

Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

A: The given problem is to prove the following statement. Given statement is |xy|=|x||y| for all real…

Q: V × (V × F) = V(V-F) – V°F. %3D

A:

Q: 7. −ydx + (x + √xy)dy = 0

A:

Q: The direchional derivative of the salan funchion at (l,-1,0) in the direction (i-j+2k) is xy- z- xyz…

A: Given that  Let f(x,y,z)=x^2y-y^2z-xyz To find the directional derivative at point

Q: Q: Using lagrange's theorem 0. 1+ x prove that

A:

Q: Sotve caleulate the thind ordere Tayloru polynemial about (o,0)

A:

Q: Prove where the following complex function is analytic

A:

Q: Find curl(zi + xj+ yk).

A: Using definition of curl of vector.