Engineering Math Essentials

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Multiple Integrals Double Integrals Definition: $\iint_R f(x,y) \,dA$ Iterated Integrals: $\int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \,dy\,dx$ $\int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y) \,dx\,dy$ Change of Order of Integration: Sketch the region of integration to determine new limits. Area: Area of region $R = \iint_R dA$ Triple Integrals Definition: $\iiint_D f(x,y,z) \,dV$ Volume: Volume of solid $D = \iiint_D dV$ Vector Calculus Vector Differentiation Gradient: For a scalar function $\phi(x,y,z)$, the gradient is $\nabla \phi = \frac{\partial \phi}{\partial x}\mathbf{i} + \frac{\partial \phi}{\partial y}\mathbf{j} + \frac{\partial \phi}{\partial z}\mathbf{k}$. Points in the direction of maximum increase of $\phi$. Magnitude is the maximum rate of increase. Normal to level surfaces $\phi(x,y,z) = C$. Directional Derivative: $D_{\mathbf{u}}\phi = \nabla \phi \cdot \mathbf{u}$, where $\mathbf{u}$ is a unit vector. Divergence: For a vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$, $\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$. Measures the outflow/inflow at a point. Solenoidal Field: If $\nabla \cdot \mathbf{F} = 0$. Curl: For a vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$, $\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$. Measures the rotation/circulation at a point. Irrotational Field: If $\nabla \times \mathbf{F} = \mathbf{0}$. Vector Integration Line Integral: $\int_C \mathbf{F} \cdot d\mathbf{r}$ or $\int_C \phi \,ds$ Surface Integral: $\iint_S \mathbf{F} \cdot d\mathbf{S}$ or $\iint_S \phi \,dS$ Volume Integral: $\iiint_D \phi \,dV$ Integral Theorems Green's Theorem: (Statement only) $\oint_C (P\,dx + Q\,dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \,dA$ Gauss's Divergence Theorem: (Statement only) $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_D (\nabla \cdot \mathbf{F}) \,dV$ Stokes' Theorem: (Statement only) $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ Ordinary Differential Equations Second and Higher Order Linear ODEs with Constant Coefficients General Form: $a_n y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_1 y' + a_0 y = g(x)$ Homogeneous Equation ($g(x)=0$): Characteristic Equation: $a_n m^n + a_{n-1} m^{n-1} + \dots + a_1 m + a_0 = 0$ Roots and Solutions: Real and distinct roots $m_1, m_2, \dots$: $y_c = C_1 e^{m_1 x} + C_2 e^{m_2 x} + \dots$ Real and repeated roots $m$ (k times): $y_c = C_1 e^{m x} + C_2 x e^{m x} + \dots + C_k x^{k-1} e^{m x}$ Complex conjugate roots $m = \alpha \pm i\beta$: $y_c = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x))$ Non-Homogeneous Equation ($g(x) \neq 0$): $y = y_c + y_p$ Method of Variation of Parameters: For $y'' + P(x)y' + Q(x)y = f(x)$, if $y_1, y_2$ are solutions to homogeneous equation. $y_p = u_1 y_1 + u_2 y_2$, where $u_1' = -\frac{y_2 f(x)}{W(y_1, y_2)}$ and $u_2' = \frac{y_1 f(x)}{W(y_1, y_2)}$ Wronskian $W(y_1, y_2) = y_1 y_2' - y_2 y_1'$ Cauchy-Legendre Equation (Euler-Cauchy) Form: $ax^2 y'' + bxy' + cy = g(x)$ Homogeneous Solution: Assume $y = x^m$. Characteristic Equation: $am(m-1) + bm + c = 0$ Roots determine solution: Real distinct $m_1, m_2$: $y_c = C_1 x^{m_1} + C_2 x^{m_2}$ Real repeated $m$: $y_c = C_1 x^m + C_2 x^m \ln|x|$ Complex conjugate $\alpha \pm i\beta$: $y_c = x^\alpha (C_1 \cos(\beta \ln|x|) + C_2 \sin(\beta \ln|x|))$ System of Simultaneous Linear ODEs with Constant Coefficients Form: $\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t)$ Homogeneous Solution ($\mathbf{f}(t) = \mathbf{0}$): Find eigenvalues $\lambda_i$ and eigenvectors $\mathbf{k}_i$ of matrix $A$. Solution: $\mathbf{x}_c = C_1 \mathbf{k}_1 e^{\lambda_1 t} + C_2 \mathbf{k}_2 e^{\lambda_2 t} + \dots$