Vector Calculus

Surface Integrals, Divergence, Curl and Potential Exercise 1 : Coordinate Planes Given three squares with an area of 4 units each. They are placed: in the x-yplane, in the x-zplane, and in the y-zplane. Determine the surface elements. Exercise 2 : Rectangle Vector Element Given a rectangle with area $a \cdot b$, determine the vector element. Exercise 3 : Flux Through Given Surface Elements Compute the flow of the vector field $\mathbf{F}(x, y, z) = (5, 3, 0)$ through the surfaces given by the respective surface elements: ...

Theory of Vector Fields Exercise 1 : Basic Operations on Vector Fields Given the vector fields: $\mathbf{F}_1 = x^2 \hat{\mathbf{z}}$ $\mathbf{F}_2 = x \hat{\mathbf{x}} + y \hat{\mathbf{y}} + z \hat{\mathbf{z}}$ $\mathbf{F}_3 = yz\hat{x} + zx\hat{y} + xy\hat{z}$ Calculate $\nabla \cdot \mathbf{F}_i$ and $\nabla \times \mathbf{F}_i$ for $i=1,2,3$. Which fields are conservative? Find scalar potentials where possible. Which fields are solenoidal? Find vector potentials where possible. Show that $\mathbf{F}_3$ can be expressed both as a gradient and a curl. Exercise 2 : Helmholtz Theorem Implications For a vector field $\mathbf{F}$ in 3D, prove the following implications: ...

Vector Algebra Exercise 1 : Distributivity of Dot and Cross Prove that the dot and cross products are distributive, using definitions and diagrams: for three coplanar vectors in the general case Exercise 2 : Cross Product Associativity? Is the cross product associative? That is, does the following hold? $$ (A \times B) \times C = A \times (B \times C) $$ Prove or provide a counterexample. Exercise 3 : BAC-CAB Identity Prove the vector identity: $A \times (B \times C) = B,(A \cdot C) - C,(A \cdot B)$ (the BAC-CAB rule) by writing both sides in component form. ...

Line, Volume and Surface Integrals Exercise 1 : Basic Line Integral For $\mathbf{v} = x^2,\hat{\mathbf{x}} + 2yz,\hat{\mathbf{y}} + y^2,\hat{\mathbf{z}}$, compute $\int_C \mathbf{v} \cdot d\mathbf{l}$ from $(0,0,0)$ to $(1,1,1)$ along: $(0,0,0) \to (1,0,0) \to (1,1,0) \to (1,1,1)$ $(0,0,0) \to (0,0,1) \to (0,1,1) \to (1,1,1)$ The straight line $x = y = z$ Evaluate $\oint \mathbf{v} \cdot d\mathbf{l}$ around the closed loop: out via 1., back via 2.. Exercise 2 : Gradient Field Verification Given $\phi = e^{xyz}$, compute $\nabla \phi$ and verify the gradient theorem: $\int_{\mathbf{a}}^{\mathbf{b}} \nabla \phi \cdot d\mathbf{l} = \phi(\mathbf{b}) - \phi(\mathbf{a})$ for $\mathbf{a} = (0,0,0)$ and $\mathbf{b} = (1,1,1)$ along two different paths. ...

Dirac Delta Function Exercise 1 : Basic Delta Integrals Evaluate the following integrals: $\displaystyle \int_{2}^{6} (3x^2 - 2x - 1) , \delta(x - 3) , dx$ $\displaystyle \int_{0}^{5} \cos x ; \delta(x - \pi) , dx$ $\displaystyle \int_{-1}^{3} x^3 , \delta(x + 1) , dx$ $\displaystyle \int_{-\infty}^{\infty} \ln(x + 3) , \delta(x + 2) , dx$ $\displaystyle \int_{0}^{\infty} e^{-x} , \delta(x - 2) , dx$ Exercise 2 : Delta with Scaled & Shifted Arguments Use $\delta(ax) = \frac{1}{|a|}\delta(x)$ and similar properties to evaluate: ...

Gradient, Divergence, Curl, and Identities Exercise 1 Calculate the directional derivative of $f(x,y,z) = x^2y + y^2z$ in the direction of $\hat{\mathbf{n}} = (\hat{\mathbf{x}} + \hat{\mathbf{y}} + \hat{\mathbf{z}})\sqrt{3}$ at point $(1,2,3)$. Exercise 2 Compute the gradient of the following scalar fields: $f(x,y,z) = x^2 + 2xy + 3z + 4$ $g(x,y,z) = \sin x \sin y \sin z$ $h(x,y,z) = e^{-5x}\sin 4y\cos 3z$ Exercise 3 For the vector field $\mathbf{F} = x^2\hat{\mathbf{x}} + 3xz^2\hat{\mathbf{y}} - 2xz\hat{\mathbf{z}}$: ...

Curvilinear Coordinates Exercise 1 : Inverting Spherical Coordinates Invert the transformation $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$ to express $(r, \theta, \phi)$ in terms of $(x, y, z)$. Exercise 2 : Coordinate Examples (Conversions) Invert the transformation $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$ to express $(r, \theta, \phi)$ in terms of $(x, y, z)$. Express the point $(x, y, z) = (2, -2, 1)$ in spherical and cylindrical coordinates. Express $(r, \theta, \phi) = (3, \pi/3, \pi/4)$ in Cartesian coordinates. Express $(s, \phi, z) = (4, 5\pi/6, -2)$ in Cartesian coordinates. Exercise 3 : Spherical Unit Vectors Express the spherical unit vectors $\hat{\mathbf{r}}$, $\hat{\boldsymbol{\theta}}$, $\hat{\boldsymbol{\phi}}$ in terms of the Cartesian unit vectors $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$, i.e., derive: ...