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- Q3]a)Describe arctanh(z) in terms of logarithms. c) Find derivative of arctanh(z). Q4] A complex function in term of polar coordinates, (r, ) is described as f(z)=u(r,0)+j v(r,0). The Cauchy-Riemann equations in polar coordinates are 1 dv ar ди dv 1ди %3D r d0 ar And the Laplace equation in polar equation in polar coordinates is 1 a2ø r or ' r2 a02 1 дФ %3D ar2 By employing these equations, show that Ø(r,0)=°cos(20) is harmonic and obtain harmonic conjugate, v(r,0) of u(r,0). The auxiliary of the harmonic conjugate is v (0,0) =0. HUAWEI Nova 2 Plus DUAL CAMERALet a rectangular a'y'z'-coordinate system be obtained by rotating a rectangular xyz-coordinate system counterclockwise about the z-axis (looking down the positive z-axis) through the angle 0 Find the 4 x'y'z'-coordinates of the point whose xyz-coordinates are (0, 3, 0). = Z || ||Find a parametric description for the curve y=4-x^2 from (-2, 0) to (2, 0) such that t=0 corresponds to (-2, 0)
- Use Green's theorem to evaluate ·ld' F. dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = x) (x = (y - In(x² + y²), 2 tan-¹1 n-¹ (z)), C is the circle (x - 2)2 + (y - 3)2 = 16 oriented counterclockwiseLet C be the curve connecting (0,0,0) to (1,2,1) to (-2,2,4) to (0,0,0) 2 | (tan (a) + 2yz) dæ + +x ) dy + xyzdz 1+ y EvaluateFind parametric equations for the tangent line to the helix with parametric equations x = 2 cos(t), y = 4 sin(t), and z = t at the point ( 0, 4, 4,7). Solution The vector equation of the helix is r(t) = (2 cos(t), 4 sin(t), t), so r'(t) = (0,4,7) is t (7²) = 2 , so by the equations x = Xo+at, y = Yo + bt, and z = Zo + ct, its parametric equations are the following. The parameter value corresponding to the point 0, 4, to the vector (x(t), y(t), z(t)) = = so the tangent vector there is r' I The tangent line is the line through (0,4, 1) ₁ paralle
- Consider the curve C shown in the attached figure, which is the intersection between surfaces S, and S2 with S;: a (z - a) = (y - a)² and Sz: x2 + 4y2 = 4a² , for a> 0. a(z – a) = (y – a)² 2' + 4y² = 4a²It is defined as group of quantities as functions of one or more independent variables. * O Curvature Parametric equation Indeterminate Forms Parametersk) Find a Harmonic Conjugate v(x, y) = _ of u(x,y) = x² -y²
- Exercise B Use the theorem on saddle-node bifurcations from the lecture! to show that i = sin(r) sin(r) + cos(r) – e" has a saddle-node bifurcation at (r*, x*) = (0,0).Let F and G be vector-valued functions such that - ₹(t) = (cos(ït), e²t−1, t² − 1), Ġ(1) = (1,1,−1), Ġ'(1) = (2,3,2), Ġ″(1) = (0,1,0) Find a vector equation of the tangent line to the graph of ₹ at (−1, e, 0).Vector 1 A) Prove that: V x (FxG) = (GV)F- (FV)G+ F(VG) - G(V.F). B) Find all of the second derivatives for f(x. y) = (3xy² + 2xy + x²) In: +y² Brie differente directia derivatives with stable gnh an eations. Find the direcional avati 220. vector in t direction of ere E) Evaluat the trde integral I need answer Only branch A (a) dx dy= (b) dyd = 24 GEL direct رسالت +vana is the unit