?Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the

Original Equilibrium In Problems 1-6, show that each system has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist. \(x^{\prime}=-2 x+3 y+x y\) \(y^{\prime}=-x+y-2 x y^{2}\) Equation Transcription: Text Transcription: x’ = -2x+3y+xy y’ = -x+y-2xy^2

Original Equilibrium In Problems 1-6, show that each system has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist. \(x^{\prime}=-y-x^{3}\) \(y^{\prime}=x-y^{3}\) Equation Transcription: Text Transcription: x' = -y - x^3 y' = x - y^3

Original Equilibrium In Problems 1-6, show that each system has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist. \(x^{\prime}=x+y+2 x y\) \(y^{\prime}=-2 x+y+y^{3}\) Equation Transcription: Text Transcription: x' = x + y + 2xy y' = -2x + y + y^3

Original Equilibrium In Problems 1-6, show that each system has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist. \(x^{\prime}=y\) \(y^{\prime}=-\sin x-y\) Equation Transcription: Text Transcription: x' = y y' = -sin x - y

Original Equilibrium In Problems 1-6, show that each system has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist. \(x^{\prime}=x+y^{2}\) \(y^{\prime}=x^{2}+y^{2}\) Equation Transcription: Text Transcription: x' = x + y^2 y' = x^2 + y^2

Original Equilibrium In Problems 1-6, show that each system has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist. \(x^{\prime}=\sin\ y\) \(y^{\prime}=-\sin\ x+y\) Equation Transcription: Text Transcription: x' = sin y y' = -sin x + y

Unusual Equilibria For each system in Problems 7-9. determine the type and stability of each real equilibrium point by calculating the Jacobian matrix at each equilibrium. \(x^{\prime}=1-x y\) \(y^{\prime}=x-y^{3}\) Equation Transcription: Text Transcription: x' = 1 - xy y' = x - y^3

Unusual Equilibria For each system in Problems 7-9. determine the type and stability of each real equilibrium point by calculating the Jacobian matrix at each equilibrium. \(x^{\prime}=x-3 y+2 x y\) \(y^{\prime}=4 x-6 y-x y\) Equation Transcription: Text Transcription: x' = x - 3y + 2xy y' = 4x - 6y - xy

Unusual Equilibria For each system in Problems 7-9. determine the type and stability of each real equilibrium point by calculating the Jacobian matrix at each equilibrium. \(x^{\prime}=4 x-x^{3}-x y^{2}\) \(y^{\prime}=4 y-x^{2} y-y^{3}\) Equation Transcription: Text Transcription: x' = 4x - x^3 - xy^2 y' = 4y - x^2 y - y^3

Linearization Completion Complete the analysis started in Example 1 by providing the details of the linearization about the point \((- 1, \ 0)\) for \(x^{\prime}=y,\ y^{\prime}=-y+x-x^{3}\). Equation Transcription: Text Transcription: (-1, 0) x' = y, y' = -y + x - x^3

Uncertainty Because a center equilibrium is stable but not asymptotically stable, nonlinear perturbation can have different outcomes. shown in Problems 11 and 12. Determine the stability of the equilibrium solutions of the strong spring \(\ddot{x}+\dot{x}+x+x^{3}=0\) Equation Transcription: Text Transcription: \ddot x + \dot x + x + x^3 = 0

Uncertainty Because a center equilibrium is stable but not asymptotically stable, nonlinear perturbation can have different outcomes. shown in Problems 11 and 12. Determine the stability of the equilibrium solutions of the weak spring \(\ddot{x}+\dot{x}+x-x^{3}=0\) Equation Transcription: Text Transcription: \ddot x + \dot x + x - x^3 = 0

Lienard Equation A generalized damped mass-spring equation, the Lienard equation, is \(\ddot{x}+p(x) \dot{x}+q(z)=0\). If \(q(0)=0\), \(\dot{q}(0)>0\), and \(p(0)>0\), show that the origin is a stable equilibrium point. Equation Transcription: Text Transcription: 2 dots x + p(x) 1dot x + q(z) = 0 q(0) = 0 q(0) > 0 p(0) > 0

Conservative Equation A second-order DE of the form \(\ddot{x}+F(x)=0\) is called a conservative differential equation. (See Sec. 4.7.) Find the equilibrium points of the conservative equation \(\ddot{x}+x-x^{2}-2 x^{3}=0\) and determine their type and stability. Equation Transcription: Text Transcription: 2 dots x + F(x) = 0 2 dots x + x - x^2 - 2x^3 = 0

Predator-Prey Equations In Sec. 2.6 we introduced the Lotka-Volterra predator-prey system \(x^{\prime}=(a-b y) x\), \(y^{\prime}=(c x-d) y\), and determined its equilibrium points \((0, \ 0)\) and \((d/c, \ a/b)\). Use the Jacobian matrix to analyze the stability around the equilibrium point \((d/c, \ a/b)\). Interpret the trajectories of this system as plotted in Fig. 2.6. 7. Equation Transcription: Text Transcription: x' = (a - by)x y'(cx - d)y (0, 0) (d/c, a/b) (d/c, a/b)

van der Pol's Equation Show that the zero solution of van derPol's equation, \(\ddot{x}-\varepsilon\left(1-x^{2}\right) \dot{x}+x=0\), is unstable for any positive value of parameter \(boldsymbol{\varepsilon}\). Equation Transcription: Text Transcription: 2 dots x - varepsilon(1 - x^2)1 dot x + x = 0 varepsilon

Damped Mass-Spring Systems The second-order linear DE \(m \ddot{x}+b \dot{x}+k x=0\) models vibrations of a mass \(m\) attached to a spring with spring constant \(k\) and damping constant \(b\). For the nonlinear variations in Problems 17-20. use your intuition to decide whether the zero solution \((x=\dot{x} \equiv 0)\) is stable or unstable. Check your intuition by transforming to a first-order system and linearizing. \(\ddot{x}+\dot{x}^{3}+x=0\) Equation Transcription: Text Transcription: m 2 dots x + b 1 dot x + kx = 0 m k b (x = 1 dot x equiv 0) 2 dots x + 1 dot x^3 + x = 0

Damped Mass-Spring Systems The second-order linear DE \(m \ddot{x}+b \dot{x}+k x=0\) models vibrations of a mass \(m\) attached to a spring with spring constant \(k\) and damping constant \(b\). For the nonlinear variations in Problems 17-20. use your intuition to decide whether the zero solution \((x=\dot{x} \equiv 0)\) is stable or unstable. Check your intuition by transforming to a first-order system and linearizing. \(\ddot{x}+\dot{x}-\dot{x}^{3}+x=0\) Equation Transcription: Text Transcription: m 2 dots x + b 1 dot x + kx = 0 m k b (x = 1 dot x equiv 0) 2 dots x + 1 dot x - 1 dot x^3 + x = 0

Damped Mass-Spring Systems The second-order linear DE \(m \ddot{x}+b \dot{x}+k x=0\) models vibrations of a mass \(m\) attached to a spring with spring constant \(k\) and damping constant \(b\). For the nonlinear variations in Problems 17-20. use your intuition to decide whether the zero solution \((x=\dot{x} \equiv 0)\) is stable or unstable. Check your intuition by transforming to a first-order system and linearizing. \(\ddot{x}+\dot{x}+\dot{x}^{3}+x=0\) Equation Transcription: Text Transcription: m 2 dots x + b 1 dot x + kx = 0 m k b (x = 1 dot x equiv 0) 2 dots x + 1 dot x + 1 dot x^3 + x = 0

Damped Mass-Spring Systems The second-order linear DE \(m \ddot{x}+b \dot{x}+k x=0\) models vibrations of a mass \(m\) attached to a spring with spring constant \(k\) and damping constant \(b\). For the nonlinear variations in Problems 17-20. use your intuition to decide whether the zero solution \((x=\dot{x} \equiv 0)\) is stable or unstable. Check your intuition by transforming to a first-order system and linearizing. \(\ddot{x}-\dot{x}+x=0\) Equation Transcription: Text Transcription: m 2 dots x + b 1 dot x + kx = 0 m k b (x = 1 dot x equiv 0) 2 dots x - 1 dot x + x = 0

Liapunov Functions An alternative approach to determining stability is the direct method of Liapunov. Liapunov assumes the existence of a positive-definite energylike function \(L(x, \ y)\) with continuous first partial derivatives. His theorem states that if \((0, \ 0)\) is an isolated equilibrium solution of \(x^{\prime}=f(x,\ y)\) and \(y^{\prime}=g(x,\ y)\), and if \(\frac{d L}{d t}=\frac{\partial L}{\partial x} \frac{d x}{d t}+\frac{\partial L}{\partial y} \frac{d y}{d t}=L_{x} x^{\prime}+L_{y} y^{\prime}\) (the derivative of \(L\) along the trajectory) is negative definite on a neighborhood of the origin, then the origin is asymptotically stable. Use Liapunov's direct method to verify the asymptotic stability of the origin for each system in Problems 21 and 22, after checking that the given function \(L\) is a legitimate Liapunov function. \(x^{\prime}=y-2 x^{3}\) \(y^{\prime}=-2 x-3 y^{5}\) \(L(x,\ y)=2x^2+y^2\) Equation Transcription: Text Transcription: L(x, y) (0, 0) x' = f(x, y) y’ = g(x, y) dL/dt = partial L/partial x dx/dt + partial L/partial y dy/dt = L_x x' + L_y y' L L x' = y - 2x^3 y' = -2x - 3y^5 L(x, y) = 2x^2 + y^2

Liapunov Functions An alternative approach to determining stability is the direct method of Liapunov. Liapunov assumes the existence of a positive-definite energylike function \(L(x, \ y)\) with continuous first partial derivatives. His theorem states that if \((0, \ 0)\) is an isolated equilibrium solution of \(x^{\prime}=f(x,\ y)\) and \(y^{\prime}=g(x,\ y)\), and if \(\frac{d L}{d t}=\frac{\partial L}{\partial x} \frac{d x}{d t}+\frac{\partial L}{\partial y} \frac{d y}{d t}=L_{x} x^{\prime}+L_{y} y^{\prime}\) (the derivative of \(L\) along the trajectory) is negative definite on a neighborhood of the origin, then the origin is asymptotically stable. Use Liapunov's direct method to verify the asymptotic stability of the origin for each system in Problems 21 and 22, after checking that the given function \(L\) is a legitimate Liapunov function. \(x^{\prime}=2 y-x^{3}\) \(y^{\prime}=-x^{3}-y^{5}\) \(L(x,\ y)=x^4+4y^2\) Equation Transcription: Text Transcription: L(x, y) (0, 0) x' = f(x, y) y’ = g(x, y) dL/dt = partial L/partial x dx/dt + partial L/partial y dy/dt = L_x x' + L_y y' L L x' = 2y - x^3 y' = -x^3 - y^5 L(x, y) = x^4 + 4y^2

A Bifurcation Point If a nonlinear system depends on a parameter \(k\) (such as a damping constant, spring constant, or chemical concentration), a critical value \(k_{0}\) where the qualitative behavior of the system changes is called a bifurcation point. Show that \(k=0\) is a bifurcation point for the system \(x^{\prime}=-x\left(y^{2}+1\right)\), (15) \(y^{\prime}=y^{2}+k\) as follows. Illustrate each part with a phase portrait. (a) Show that (15) has two equilibrium points for \(k<0\). (b) Show that (15) has one equilibrium point for \(k=0\). (c) Show that (15) has no equilibrium points for \(k>0\). (d) Calculate the linearization about the equilibrium point for \(k=0\). Relate the phase portraits for (b) and (d). Equation Transcription: Text Transcription: k k_0 k = 0 x' = -x(y^2 + 1) y' = y^2 + k k < 0 k = 0 k > 0 k = 0

Computer Lab: Trajectories Rewrite the second-order equation in each of Problems 24-27 as a first-order system with \(x^{\prime}=y\). Use appropriate software to sketch trajectories using the direction field for \(d y / d x=y^{\prime} / x^{\prime}\). Compare with behaviors of the linearized systems (see Chapter 4 ), and explain what is different and why. \(x^{\prime\prime}+x\ \sin\ x=0\) Equation Transcription: Text Transcription: x' = y dy/dx = y'/x' x'' + x sin x = 0

Computer Lab: Trajectories Rewrite the second-order equation in each of Problems 24-27 as a first-order system with \(x^{\prime}=y\). Use appropriate software to sketch trajectories using the direction field for \(d y / d x=y^{\prime} / x^{\prime}\). Compare with behaviors of the linearized systems (see Chapter 4 ), and explain what is different and why. \(x^{\prime \prime}+x-0.1\left(x^{2}+2 x^{3}\right)=0\) Equation Transcription: Text Transcription: x' = y dy/dx = y'/x' x'' + x - 0.1(x^2 + 2x^3) = 0

Computer Lab: Trajectories Rewrite the second-order equation in each of Problems 24-27 as a first-order system with \(x^{\prime}=y\). Use appropriate software to sketch trajectories using the direction field for \(d y / d x=y^{\prime} / x^{\prime}\). Compare with behaviors of the linearized systems (see Chapter 4 ), and explain what is different and why. \(x^{\prime \prime}-\left(1-x^{2}\right) x^{\prime}+x=0\) Equation Transcription: Text Transcription: x' = y dy/dx = y'/x' x'' - (1 - x^2)x' + x = 0

Computer Lab: Trajectories Rewrite the second-order equation in each of Problems 24-27 as a first-order system with \(x^{\prime}=y\). Use appropriate software to sketch trajectories using the direction field for \(d y / d x=y^{\prime} / x^{\prime}\). Compare with behaviors of the linearized systems (see Chapter 4 ), and explain what is different and why. \(x^{\prime \prime}+x-0.25 x^{2}=0\) Equation Transcription: Text Transcription: x' = y dy/dx = y'/x' x'' + x - 0.25x^2 = 0

Computer Lab: Competition Work IDE Lab 22 to get a visceral feel for how changing parameters affects the location and character of the equilibria. This system was discussed in detail in Sec. 2.6.

Suggested Journal Entry I Consider the tangent line linearization \(L(x)\) to the graph of a function \(f(x)\) of one variable, and discuss its relative predictive value for the behavior of \(f\) in the cases \(L^{\prime}\left(x_0\right)>0,\ L^{\prime}\left(x_0\right)=0\), and \(L^{\prime}\left(x_{0}\right)<0\). Can you draw an analogy to the linearization of an autonomous system of DEs? Equation Transcription: Text Transcription: L(x) f(x) f L'(x0) > 0 L'(x0) = 0 L'(x0) < 0

Suggested Journal Entry II Summarize the relationship between a nonlinear system and its linearization at an equilibrium point, both geometrically and in regard to stability.

Complex Plane Plot the following complex numbers in the complex plane. (a) \(3+3 i\) (b) \(4i\) (c) \(2\) (d) \(1-i\) Equation Transcription: Text Transcription: 3+3i 4i 2 1-i

Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\) Equation Transcription: Text Transcription: (2+3i)(4-i) (2+3i)(1+i) 1/1+i 2+i/3+i

Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1pi e^(2+pi i/4)

Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2i -i -1-i -2+3i e2i 2+i/1+i

Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcription: F(t)=e^(1-i)t F(t)=e^3it F(t)=e^(2+3i)t

Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2) e^pi i e^-pi i

Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\) Equation Transcription: Text Transcription: (2+3i)(4-i) (2+3i)(1+i) 1/1+i 2+i/3+i

Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1pi e^(2+pi i/4)

Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2i -i -1-i -2+3i e2i 2+i/1+i

Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcription: F(t)=e^(1-i)t F(t)=e^3it F(t)=e^(2+3i)t

Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2) e^pi i e^-pi i

Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\) Equation Transcription: Text Transcription: (2+3i)(4-i) (2+3i)(1+i) 1/1+i 2+i/3+i

Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1pi e^(2+pi i/4)

Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2i -i -1-i -2+3i e2i 2+i/1+i

Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcription: F(t)=e^(1-i)t F(t)=e^3it F(t)=e^(2+3i)t

Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2) e^pi i e^-pi i

Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\) Equation Transcription: Text Transcription: (2+3i)(4-i) (2+3i)(1+i) 1/1+i 2+i/3+i

Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1pi e^(2+pi i/4)

Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2i -i -1-i -2+3i e2i 2+i/1+i

Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcription: F(t)=e^(1-i)t F(t)=e^3it F(t)=e^(2+3i)t

Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2) e^pi i e^-pi i

Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\) Equation Transcription: Text Transcription: (2+3i)(4-i) (2+3i)(1+i) 1/1+i 2+i/3+i

Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1pi e^(2+pi i/4)

Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2i -i -1-i -2+3i e2i 2+i/1+i

Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcription: F(t)=e^(1-i)t F(t)=e^3it F(t)=e^(2+3i)t

Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2) e^pi i e^-pi i

Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\) Equation Transcription: Text Transcription: (2+3i)(4-i) (2+3i)(1+i) 1/1+i 2+i/3+i

Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1pi e^(2+pi i/4)

Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2i -i -1-i -2+3i e2i 2+i/1+i

Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Derivatives of Complex Functions Find the derivatives F’(t) and F’’(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcription: F(t)=e^(1-i)t F(t)=e^3it F(t)=e^(2+3i)t

Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2) e^pi i e^-pi i

Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Using DeMoivre's Formula Use formula (8) to find the general solutions for the DE's in Problems 13-14. \(\frac{d^{3} y}{d t^{3}}+y=0\) ________________ Equation Transcription: Text Transcription: d^3y/dt^3+y=0

Using DeMoivre's Formula Use formula (8) to find the general solutions for the DE's in Problems 13-14. \(\frac{d^{4} y}{d t^{4}}+81 y=0\) ________________ Equation Transcription: Text Transcription: d^4y/dt^4+81y=0

Coordinate Map Let V be a vector space with basis \(B=\left\{\bar{b}_{1}, \bar{b}_{2}, \ldots \bar{b}_{n}\right\}\). Show that the coordinate map \([]_{B}: V \rightarrow \mathbb{R}^{n}\)is a isomorphism. (You need to show three things: linearity, injectivity and surjectivity.) ________________ Equation Transcription: ?n Text Transcription: B={bar b_1, bar b_2,...bar b_n} []_B: V rightarrow double R^n

Isomorphisms List at least three vector spaces that are isomorphic to M22(?).

Isomorphism Subtleties Explain why M12(?)is not a subspace of M22(?). Show. however, lhat it is isomorphic to a subspace of M22(?) by finding the subspace and the isomorphism.

Isomorphisms Have Inverses Let \(T: V \rightarrow W\) be an isomorphism. Prove that the inverse map T-1 exists and is also an isomorphism. (You must define T-1 and show that it is an injective and surjective linear transformation from W to V.) ________________ Equation Transcription: Text Transcription: T:V rightarrow W

Composition of Isomorphisms Let \(T: V \rightarrow W\) be an isomorphism between vector spaces. Prove that if \(L: W \rightarrow U\) is an isomorphism between vector spaces. then the composition \(L \circ T: V \rightarrow U\) is also an isomorphism. ________________ Equation Transcription: Text Transcription: T:V rightarrow W L:W rightarrow U L circle T:V rightarrow U

Isomorphisms and Bases Let \(T: V \rightarrow W\) be an isomorphism between vector spaces. Use the properties of isomorphisms to prove that if \(\left\{\bar{b}_{1}, \bar{b}_{2}, \ldots \bar{b}_{n}\right\}\) is a basis for V. then { T(b1), T(b2),..., T(bn)} I is a basis for W. ________________ Equation Transcription: Text Transcription: T:V rightarrow W {bar b_1, bar b_2,...bar b_n}

Associated Matrices In Problems 7-11, find the matrix MB associated with the linear transformation \(T: V \rightarrow W\) from basis B for V to basis C for W. V = ?2, W = ?3, \(T(x,y) = (2x-y, x , y)\), where B and C are the standard bases for ?2 and ?3 , respectively. ________________ Equation Transcription: Text Transcription: T:V rightarrow W T(x,y) = (2x-y, x , y)

Associated Matrices In Problems 7-11, find the matrix MB associated with the linear transformation \(T: V \rightarrow W\) from basis B for V to basis C for W. V = P2, W = ?3, \(T\left(a t^{2}+b t+c\right)=\left[\begin{array}{l} a-b \\a \\ 2 c \end{array}\right] \), where \(B=\left\{t^{2}, t, 1\right\}\) and \(C=\left\{\bar{e}_{1}, \bar{e}_{2}, \bar{e}_{3}\right\}\). ________________ Equation Transcription: [] Text Transcription: T:V rightarrow W T(at^2+bt+c)=[a-b a 2c] B={t^2, t, 1} C={bar e_1, bar e_2, bar e_3}

Associated Matrices In Problems 7-11, find the matrix MB associated with the linear transformation \(T: V \rightarrow W\) from basis B for V to basis C for W. V = M22(?), W = M22(?), T(A) = A + AT and \(B=C=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} \). HINT: Note that \(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]_{c}=\left[\begin{array}{l} a \\ b \\ c \\ d \end{array}\right] \), so MB will be a 4 x 4 matrix. ________________ Equation Transcription: . Text Transcription: T:V rightarrow W B=C={[1 0 0 0], [0 1 0 0], [0 0 1 0], [0 0 0 1]}

Associated Matrices In Problems 7-11, find the matrix MB associated with the linear transformation \(T: V \rightarrow W\) from basis B for V to basis C for W. V = M22(?), W = M22(?), T(A) = \(\left[\begin{array}{cc} \operatorname{Tr} A & 0 \\ 0 & \operatorname{Tr} A \end{array}\right] \) and \(B=C=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} \). See hint for Problem 9. ________________ Equation Transcription: . Text Transcription: T:V rightarrow W [TrA 0 0 TrA] B=C={[1 0 0 0], [0 1 0 0], [0 0 1 0], [0 0 0 1]}

Associated Matrices In Problems 7-11, find the matrix MB associated with the linear transformation \(T: V \rightarrow W\) from basis B for V to basis C for W. V = W is the solution space for \(x^{\prime \prime}+4 x^{\prime}+4 x=0, T(f)=f^{\prime}-f\) and \(B=C=\left\{e^{-2 t}, t e^{-2 t}\right\}\) ________________ Equation Transcription: , Text Transcription: T:V rightarrow W x''+4x'+4x=0, T(f) = f' - f B=C={e^-2t, te^-2t}

Changing Bases In Problems 12-14, determine the matrix associated with a change of bases from basis B to basis C for the given vector space V. V = M21(?), \(B=\left\{\left[\begin{array}{l} 1 \\ 1 \end{array}\right],\left[\begin{array}{l} 3 \\ 0 \end{array}\right]\right\}, \quad C=\left\{\left[\begin{array}{r} -1 \\ 1 \end{array}\right],\left[\begin{array}{l} 0 \\ 2 \end{array}\right]\right\} \) ________________ Equation Transcription: Text Transcription: B={[1 1], [3 0]}, C= {[-1 1], [0 2]}

Changing Bases In Problems 12-14, determine the matrix associated with a change of bases from basis B to basis C for the given vector space V. V = P3, \(B=\left\{t^{3}, t^{2}, t, 1\right\}, C=\left\{2 t, t^{3}, t-t^{2}, 5\right\}\). ________________ Equation Transcription: Text Transcription: B={t^3,t^2, t, 1}, C={2t, t^3, t-t^2,5}

Changing Bases In Problems 12-14, determine the matrix associated with a change of bases from basis B to basis C for the given vector space V. V = M22(?), \(\begin{array}{l} B=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\}, \\ C=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\right\} \end{array} \) ________________ Equation Transcription: Text Transcription: B={[1 0 0 0], [0 1 0 0], [0 0 1 0], [0 0 01]} C={[1 0 0 0], [1 1 0 0], [1 1 1 0], [1 1 1 1]}

Associated Matrix Again Return to Problem 8 but replace the C basis for ?3 by \(D=\left\{\bar{e}_{1}, \bar{e}_{2}-\bar{e}_{2}, 5 \bar{e}_{3}+\bar{e}_{1}\right\}\). Determine the matrix \(M_{B}^{*}\) associated with a change from basis B to basis D. HINT: Start with the fact that \(\left[T\left(t^{2}\right)\right]_{D}=[(1,1,0)]_{D}\) \(=\left[\delta_{1}\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right]+\delta_{2}\left[\begin{array}{r}1 \\ -1 \\ 0 \end{array}\right]+\delta_{3}\left[\begin{array}{l} 1 \\ 0 \\ 5 \end{array}\right]\right]_{D} \) \(=\left[\begin{array}{l} \delta_{1} \\ \delta_{2} \\ \delta_{3} \end{array}\right] \) ________________ Equation Transcription: Text Transcription: D={bar e_1,bar e_1-bar e_2, 5 bar e_3+bar e_1} M^*_B [T(t^2)]_D=[(1,1,0)]_D =[delta_1 [1 0 0] + delta_2[1 -1 0] + delta_3[1 0 5]]_D =[delta_1 delta_2 delta_3]]

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{1}{x(x-1)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 1/x(x - 1)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{1}{(x+2)(x-1)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 1/(x + 2)(x - 1)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{x}{(x+1)(x+2)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 x/(x + 1)(x + 2)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{x}{\left(x^{2}+1\right)(x-1)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 x/(x^2 + 1)(x - 1)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{4}{x^{2}\left(x^{2}+4\right)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 4/x^2 (x^2 + 4)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{3}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 3/(x^2 + 1)(x^2 + 4)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{7 x-1}{(x+1)(x+2)(x-3)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 7x - 1/(x + 1)(x + 2)(x - 3)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{x^{2}-2}{x(x+7)(x+1)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 x^2 - 2/x(x + 7)(x + 1)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{x^{2}+9 x+2}{(x-1)^{2}(x+3)}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 x^2 + 9x + 2/(x - 1)^2 (x + 3)

Practice Makes Perfect Resolve the rational fraction in each of \(\text { Problems } \ 1-10 \) into its partial fraction decomposition. \(\frac{x^{2}+1}{x^{3}-2 x^{2}-8 x}\) Equation Transcription: Problems 1-10 Text Transcription: Problems 1-10 x^2 + 1/x^3 - 2x^2 - 8x

Multiplying Associated Matrices Return to Problems 15 and 8, with bases \(B=\left\{t^{2}, t, 1\right\}\), \(C=\left\{\bar{e}_{1}, \bar{e}_{2}, \bar{e}_{3}\right\}\), \(D=\left\{\bar{e}_{1}, \bar{e}_{1}-\bar{e}_{2}, 5 \bar{e}_{3}+\bar{e}_{1}\right\}\) (a) Find \(M_{C}^{*}\) for the change of basis from C to D. (b) Verify that \(M_{B}^{*}\) from Problem 15 can be calculated with \(M_{B}^{*}=M_{C}^{*} M_{B}\), where \(M_{C}^{*}\) is from (a) and \(M_{B}^{*}\) is from Problem 8. Explain why this should be so. ________________ Equation Transcription: , = Text Transcription: B={t^2,t,1}, C={bar e_1,bar e_2,bar e_3} D={bar e_1,bar e_1-bar e_2, 5 bar e_3+bar e_1} M^*_C M^*_B M^*B_=M^*_C M_B