Partial Differentiation: Part 3

Advanced Numerical Problems on Partial Differentiation

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Problem 1: Laplace’s Equation

Problem: Verify that \( u = x^3 - 3xy^2 \) satisfies Laplace’s equation.

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\[ u_x = 3x^2 - 3y^2, \quad u_y = -6xy \] \[ u_{xx} = 6x, \quad u_{yy} = -6x \] \[ u_{xx} + u_{yy} = 6x - 6x = 0 \]

Hence verified.

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Problem 2: Total Differential

Problem: If \( z = e^{xy}\sin(x+y) \), find total differential \( dz \).

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\[ dz = z_x dx + z_y dy \] \[ z_x = e^{xy}[y\sin(x+y)+\cos(x+y)] \] \[ z_y = e^{xy}[x\sin(x+y)+\cos(x+y)] \] \[ dz = e^{xy}[y\sin(x+y)+\cos(x+y)]dx \] \[ + e^{xy}[x\sin(x+y)+\cos(x+y)]dy \]

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Problem 3: Directional Derivative

Problem: Find directional derivative of \( \phi=x^2y+yz^3 \) at (1,-1,1) in direction \(2\mathbf{i}-\mathbf{j}+2\mathbf{k}\).

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\[ \nabla \phi = (2xy)\mathbf{i} + (x^2+z^3)\mathbf{j} + (3yz^2)\mathbf{k} \] At (1,-1,1): \[ \nabla \phi = (-2)\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \] Direction vector magnitude: \[ |\vec{a}| = 3 \] Unit vector: \[ \hat{a}=\frac{1}{3}(2,-1,2) \] Directional derivative: \[ \nabla\phi \cdot \hat{a} = -4 \]

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Problem 4: Taylor Expansion

Problem: Expand \( f(x,y)=e^x\cos y \) about (0,0) up to second order.

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\[ f(0,0)=1 \] \[ f_x(0,0)=1, \quad f_y(0,0)=0 \] \[ f_{xx}=1,\quad f_{yy}=-1,\quad f_{xy}=0 \] \[ f(x,y)=1+x+\frac{1}{2}(x^2-y^2) \]

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Problem 5: Jacobian

Problem: If \( x=r\cos\theta, y=r\sin\theta \), find Jacobian \( \frac{\partial(x,y)}{\partial(r,\theta)} \).

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\[ J= \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} \] \[ = r(\cos^2\theta+\sin^2\theta)=r \]

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Problem 6: Lagrange Multiplier

Problem: Find extrema of \( f=x^2+y^2 \) subject to \( x+y=1 \).

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\[ \nabla f=(2x,2y), \quad \nabla g=(1,1) \] \[ (2x,2y)=\lambda(1,1) \Rightarrow x=y \] Using constraint: \[ 2x=1 \Rightarrow x=y=\frac12 \] Minimum value: \[ f=\frac12 \]

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Problem 7: Implicit Differentiation

Problem: If \( x^2+y^2+xy=7 \), find \( \frac{dy}{dx} \).

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\[ 2x+2y\frac{dy}{dx}+x\frac{dy}{dx}+y=0 \] \[ (2y+x)\frac{dy}{dx}=-(2x+y) \] \[ \frac{dy}{dx}=-\frac{2x+y}{2y+x} \]

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Problem 8: Harmonic Function

Problem: Show \( u=\ln(x^2+y^2) \) is harmonic.

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\[ u_x=\frac{2x}{x^2+y^2} \] \[ u_{xx}=\frac{2(y^2-x^2)}{(x^2+y^2)^2} \] \[ u_{yy}=\frac{2(x^2-y^2)}{(x^2+y^2)^2} \] \[ u_{xx}+u_{yy}=0 \] Hence harmonic.

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Problem 9: Euler’s Theorem

Problem: Verify Euler’s theorem for \( u=(x^2+y^2)^{3/2} \).

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\[ u_x=3x(x^2+y^2)^{1/2} \] \[ u_y=3y(x^2+y^2)^{1/2} \] \[ xu_x+yu_y=3(x^2+y^2)^{3/2}=3u \]

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Problem 10: Maximum Directional Derivative

Problem: Find maximum rate of increase of \( f=xye^z \) at (1,1,0).

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\[ \nabla f= (ye^z)\mathbf{i} + (xe^z)\mathbf{j} + (xye^z)\mathbf{k} \] At (1,1,0): \[ \nabla f=(1,1,1) \] Maximum rate: \[ |\nabla f|=\sqrt3 \] Direction: along gradient vector.